Inclusion Criteria:

          -  Patients must have histologically or cytologically confirmed head and neck squamous
             cell cancer (HNSCC) including paranasal sinus cancers but excluding nasopharyngeal
             carcinomas

          -  Clinical staged III or IV HNSCC that is not amenable to surgical resection

          -  Carcinoma of the neck of unknown primary site origin (regardless of HPV/p16 status) is
             eligible

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky >= 60%)

          -  Life expectancy of greater than 3 months

          -  Patients must have measurable disease, defined as at least one lesion that can be
             accurately measured in at least one dimension (longest diameter to be recorded for
             non-nodal lesions and short axis for nodal lesions) as >= 20 mm (>= 2 cm) with
             conventional techniques or as >= 10 mm (>= 1 cm) with spiral computed tomography (CT)
             scan, magnetic resonance imaging (MRI), or calipers by clinical exam

          -  Leukocytes >= 3,000/mcL

          -  Absolute neutrophil count >= 1,500/mcL

          -  Platelets >= 100,000/mcL

          -  Total bilirubin within normal institutional limits

          -  Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT])
             =< 2.5 x institutional upper limit of normal

          -  Creatinine within normal institutional limits OR creatinine clearance >= 60
             mL/min/1.73 m^2 for patients with creatinine levels above institutional normal

          -  The effects of M6620 (VX-970, berzosertib) on the developing human fetus are unknown;
             for this reason and because DNA-damage response (DDR) inhibitors as well as other
             therapeutic agents used in this trial may have teratogenic potential, women of
             child-bearing potential and men must agree to use adequate contraception (hormonal or
             barrier method of birth control; abstinence) prior to study entry and for the duration
             of study participation; should a woman become pregnant or suspect she is pregnant
             while she or her partner is participating in this study, she should inform her
             treating physician immediately; men treated or enrolled on this protocol must also
             agree to use adequate contraception prior to the study, for the duration of study
             participation, and for 6 months after completion of M6620 (VX-970, berzosertib)
             administration

          -  Ability to understand and the willingness to sign a written informed consent document

          -  Women of childbearing potential who are sexually active should be willing and able to
             use medically acceptable forms of contraception throughout the treatment phase of the
             trial and for up to 6 months following the last administration of study treatment; men
             who are sexually active must be willing and able to use medically acceptable forms of
             contraception throughout the treatment phase of the trial and for 6 months after
             completion of M6620 (VX-970, berzosertib) administration

        Exclusion Criteria:

          -  Patients with nasopharyngeal carcinoma, skin squamous cell carcinoma (SCC), and
             salivary gland carcinomas are not eligible

          -  Patients who are receiving adjuvant chemoradiation after surgical resection of the
             primary site of disease

          -  Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for
             nitrosoureas or mitomycin C) prior to entering the study or those who have not
             recovered from adverse events due to agents administered more than 4 weeks earlier

          -  Patients who are receiving any other investigational agents

          -  Patients on tacrolimus or any other immunosuppressants with significant interaction
             with cisplatin

          -  Patient who requires live vaccine administration

          -  Patients with known brain metastases should be excluded from this clinical trial
             because of their poor prognosis and because they often develop progressive neurologic
             dysfunction that would confound the evaluation of neurologic and other adverse events

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to M6620 (VX-970, berzosertib) or cisplatin

          -  Prior systemic chemotherapy for the current cancer (prior chemotherapy for a different
             cancer is allowed)

          -  Prior receipt of radiotherapy that would result in overlap of the new and old
             radiation therapy fields

          -  Uncontrolled intercurrent illness including, but not limited to:

               -  Ongoing or active infection requiring intravenous antibiotics at the time of
                  treatment initiation

               -  Symptomatic congestive heart failure (requiring hospital stay within the last 6
                  months)

               -  Myocardial infarction within the last 6 months

               -  Unstable angina pectoris, cardiac arrhythmia

               -  Psychiatric illness/social situations that would limit compliance with study
                  requirements

          -  Pregnant women are excluded from this study because M6620 (VX-970, berzosertib) as a
             DNA-damage response (DDR) inhibitor may have the potential for teratogenic or
             abortifacient effects; because there is an unknown but potential risk for adverse
             events in nursing infants secondary to treatment of the mother with M6620 (VX-970,
             berzosertib), breastfeeding should be discontinued if the mother is treated with M6620
             (VX-970, berzosertib); these potential risks may also apply to other agents used in
             this study

          -  Human immunodeficiency virus (HIV)-positive patients with well-controlled disease, as
             determined by CD4 count and viral load, who are on antiretroviral therapy that does
             not contain a strong inducer or inhibitor of CYP3A4 are allowed on trial; HIV-positive
             patients on combination antiretroviral therapy with strong inducers or inhibitors of
             CYP3A4 are ineligible because of the potential for pharmacokinetic interactions;
             patients with poorly controlled HIV are not eligible due to the increased risk of
             lethal infections when treated with marrow-suppressive therapy

          -  Definitive clinical or radiographic evidence of distant metastasis or adenopathy below
             the clavicles

          -  M6620 (VX-970, berzosertib) is primarily metabolized by CYP3A4; therefore, concomitant
             administration with strong inhibitors or inducers of CYP3A4 should be avoided; because
             the lists of these agents are constantly changing, it is important to regularly
             consult a frequently-updated medical reference for a list of drugs to avoid or
             minimize use of; as part of the enrollment/informed consent procedures, the patient
             will be counseled on the risk of interactions with other agents, and what to do if new
             medications need to be prescribed or if the patient is considering a new
             over-the-counter medicine or herbal product