Inclusion Criteria:

        Patients must fulfill each of the following requirements:

          1. Must have a histologically or cytologically confirmed advanced solid tumor that meets
             the following additional specifications

               1. Monotherapy Part A (Dose Escalation) advanced solid tumor that is metastatic or
                  unresectable and recurrent and /or refractory to available therapy.

               2. Combination Part B (Dose Escalation)- advanced solid tumors that are (a)
                  metastatic or unresectable and recurrent and/or refractory to available therapy;
                  (b) a condition for which pembrolizumab is an approved treatment: and (c) in
                  patients who either have shown primary or acquired resistance to immune
                  checkpoint inhibitors (ICIs)

               3. Combination Arm C (Dose Expansion), Phase 2 - Advanced HNSCC tumors of the
                  oropharynx, oral cavity, hypopharynx, larynx, lip, or sinus that are (a)
                  metastatic or unresectable, and recurrent and/or refractory to available therapy,
                  (b) in patients who have been treated with pembrolizumab or other PD-1 or PD-L1
                  inhibitors in monotherapy, and (c) who have subsequently shown primary or
                  acquired resistance to ICIs.

             For enrollment in both arms:

          2. Must be ≥ 18 years of age

          3. Should have all side effects of any prior therapy or procedures for any medical
             condition recovered to CTCAE ≤ Grade 1 (except alopecia).

          4. Must have at least 1 measurable lesion by computed tomography or magnetic resonance
             imaging per RECIST v1.1.

          5. Must have a life expectancy ≥ 3 to 6 months.

          6. Female patients of childbearing age and women < 12 months since the onset of
             menopause, except those who have been surgically sterilized (tubal ligation) or whose
             sexual partner(s) is surgically sterilized (vasectomy), must agree to use acceptable
             contraceptive methods for the duration of the study and for 9 months after the date of
             their last DSP-0509 infusion. If employing contraception, 2 of the following
             precautions must be used: birth control pill, vaginal diaphragm, intrauterine system
             or device, condom or vaginal spermicide. Female patients who are postmenopausal are
             defined as those with an absence of menses for ≥ 12 consecutive months. Male patients
             must be surgically sterilized (vasectomy) or their female sexual partner(s) must be
             surgically sterilized (tubal ligation) to avoid using contraception. If they do not
             meet this criterion, then male patients or must agree to use a condom as well as one
             of the acceptable contraceptive methods listed above with their female partner(s) who
             meets the criteria of either being of childbearing age or is < 12 months since the
             onset of menopause. Male patients and their female partner(s) must agree to use
             acceptable contraception methods for the duration of time the male patient is on the
             study and for 9 months after the date of his last DSP 0509 infusion.

          7. Females of childbearing potential must have a negative serum pregnancy test.

          8. Must have an Eastern Cooperative Oncology Group performance status of 0 to 1.

          9. Must have adequate coagulation function at Screening as determined by:

               1. Prothrombin international normalized ratio < 1.5.

               2. Partial thromboplastin time < 1.5 times the upper limit of normal (ULN).

         10. Must have adequate hematologic function at Screening as determined by:

               1. White blood cell (WBC) count ≥ 3,000/microliter.

               2. Absolute neutrophil count (ANC) ≥ 1,500/microliter (patient may not use
                  granulocyte colony stimulating factor or granulocyte-macrophage colony
                  stimulating factor to achieve these WBC and ANC levels).

               3. Platelet count ≥ 100 × 103/microliter.

               4. Hemoglobin (Hgb) ≥ 9.0 g/dL (may not transfuse or use erythropoietin to obtain
                  this Hgb level).

         11. Must have adequate renal and hepatic function at Screening as determined by:

               1. Serum creatinine < 2.0 mg/dL or < 1.5 times the ULN, whichever is lower.

               2. Total bilirubin ≤ 1.5 mg/dL or < 1.5 times the ULN, whichever is lower (or ≤ 2.0
                  mg/dL for patients with known Gilbert syndrome).

               3. Aspartate aminotransferase ≤ 2.5 times ULN (≤ 5 times ULN for patients with liver
                  metastases).

               4. Alanine aminotransferase ≤ 2.5 times ULN (≤ 5 times ULN for patients with liver
                  metastases).

         12. Must be able to attend study visits as required by the protocol.

         13. Prior to the first DSP-0509 infusion, the patient must be able to provide tumor tissue
             for baseline studies either as (a) a block of archival tissue sufficient to provide
             the required number of slides (b) a sufficient number of fixed, unstained slides of
             archival tissue or (c) consent to undergo tumor biopsy to acquire sufficient tumor
             tissue. (Sites need to refer to the current version of the "Sample Collection &
             Shipment Instructions Manual" to determine how many slides are required for each
             patient as these numbers vary based on (a) the study Arm/Part in which the patient is
             enrolled and (b) whether the patient consented to optional future testing).

             In addition to the above criteria, patients must meet the following criteria to be
             eligible to enroll in Combination Arm C:

         14. Have at least one accessible tumor for biopsy. This accessible lesion must be
             considered as non-measurable per RECIST criteria, v1.1.

         15. Be platinum refractory, PD-1 or PD-L1 exposed, and have no more than 3 lines of prior
             therapy for advanced/metastatic disease

         16. Have a known status of PD-L1 combined positive score (CPS)

         17. Have a known HPV status

        Exclusion Criteria:

        Patients with any of the following will be excluded from the study:

        For enrollment in both arms:

          1. Has received prior therapy with a TLR agonist, excluding a topical TLR agonist.

          2. Has received anticancer chemotherapy (including molecular-targeted drugs),
             radiotherapy, immunotherapy (eg, vaccines or cytokines), or investigational agents
             within the 3 weeks before the first dose of DSP-0509. Local palliative radiotherapy is
             permitted3. Receives concurrent systemic (oral or IV) steroid therapy > 10 mg
             prednisone daily or its equivalent for an underlying condition.

        4. Not fully recovered from major surgery before the first dose of DSP-0509. 5. Has central
        nervous system (CNS) metastases (including leptomeningeal metastases, spinal metastases) or
        CNS primary tumors, eg, glioblastoma.

        6. Has a history of seizures other than isolated febrile seizure in childhood; has a
        history of a cerebrovascular accident or transient ischemic attack less than 6 months ago.

        7. Has effusions (pleural, pericardial, or ascites) requiring drainage. 8. Has a
        neurodegenerative disease, eg, motor neuron disease, Parkinson disease, Alzheimer disease,
        Huntington disease.

        9. Has retinal detachment, ulcerative keratitis, uveitis, Vogt-Koyanagi-Harada syndrome,
        choroidal neovascularization, retinopathy/retinitis, thyroid-associated orbitopathy,
        idiopathic orbital inflammation, diabetic retinopathy, ischemic retinopathy including
        glaucoma-associated retinopathy, retinal vein thrombosis, or a non-healing ocular or
        ophthalmic disease.

        10. Has a fever ≥ 38°C within 3 days before the first dose of study treatment.

        11. Has interstitial lung disease or active noninfectious pneumonitis. 12. Has a history of
        active autoimmune or immunologic disorder requiring immunosuppression with steroids or
        other immunosuppressive agents (eg, azathioprine, cyclosporine A) except for patients with
        isolated vitiligo, resolved childhood asthma or atopic dermatitis, controlled
        hypoadrenalism or hypopituitarism, and euthyroid patients with a history of Grave disease.
        Patients with controlled hyperthyroidism must be negative for thyroglobulin, thyroid
        peroxidase antibodies, and thyroid-stimulating immunoglobulin before study drug
        administration.

        13. Has a known hypersensitivity to a component of the protocol therapy, DSP-0509, or
        another pyrimidine.

        14. Has a history of another primary cancer within the 5 years before enrollment except for
        the following: non-melanoma skin cancer, cervical carcinoma in situ, superficial bladder
        cancer, or other nonmetastatic carcinoma that has been in complete remission without
        treatment for more than 5 years.

        15. Has abnormal ECGs that are clinically significant, such as QT prolongation (QTc > 480
        msec).

        16. In the opinion of the treating Investigator, has any concurrent conditions that could
        pose an undue medical hazard or interfere with the interpretation of the study results;
        these conditions include, but are not limited to ongoing or active infection, clinically
        significant non-healing or healing wounds, concurrent congestive heart failure (New York
        Heart Association Functional Classification Class II, III or IV), concurrent unstable
        angina, concurrent cardiac arrhythmia requiring treatment (excluding asymptomatic atrial
        fibrillation), recent (within the prior 12 months) myocardial infarction, acute coronary
        syndrome within the previous 12 months, significant pulmonary disease (shortness of breath
        at rest or on mild exertion) for example due concurrent severe obstructive pulmonary
        disease, concurrent hypertension requiring more than 2 medications for adequate control, or
        diabetes mellitus with more than 2 episodes of ketoacidosis in the prior 12 months.

        17. Has an ejection fraction of 50% or less based on a MUGA scan or ECHO. 18. Has the
        presence of a known active acute or chronic infection including human immunodeficiency
        virus as determined by enzyme-linked immunosorbent assay and confirmed by Western blot; and
        hepatitis B virus and hepatitis C virus as determined by hepatitis B surface antigen and
        hepatitis C serology.

        19. Has a cognitive, psychological, or psychosocial impediment that would impair the
        ability of the patient to receive therapy according to the protocol or adversely affect the
        ability of the patient to comply with the informed consent process, protocol, or
        protocol-required visits and procedures.

        20. Receives concurrent strong inhibitors of cytochrome P450 2C8. 21. Receives concurrent
        inhibitors of organic anion transporting peptide (OATP)1B1 and OATP1B3.

        22. Is pregnant or breastfeeding. 23. Has active neurological or inflammatory or auto
        immune disorders (e.g. Guillain-Barre Syndrome, Amyotrophic Lateral Sclerosis)

        The following exclusion applies only to enrollment in Combination arms Part B & C:

        24. Has a history of immune-related adverse events (irAEs) resulting in permanent
        discontinuation of ICI treatment.