Inclusion Criteria:

          -  Patient age 0-75 years at the time of enrollment. Initially only patients who are >=
             16 years old will receive HA-1-TCR T cell infusions on the protocol. Younger patients
             may be screened, enrolled in the protocol and monitored for relapse but will not be
             eligible for infusion until at least one patient >=16 years old has been treated and
             discussed with the Food and Drug Administration (FDA)

          -  Patients must express HLA-A*0201

          -  Patients must have the HA-1(H) genotype (RS_1801284: A/G, A/A)

          -  Patients must have an adult donor for HCT who is adequately HLA matched by
             institutional standards (includes HLA-matched related or unrelated donors, and
             HLA-mismatched family donors, including haploidentical donors) and is either:

               -  HLA-A*0201 positive and HA-1(H) negative (RS_1801284: G/G) or

               -  HLA-A*0201 negative

          -  Patients who are currently undergoing or who previously underwent allogeneic HCT for

               -  Acute myeloid leukemia (AML) of any subtype

               -  Acute lymphoid leukemia (ALL) of any subtype

               -  Mixed phenotype/undifferentiated/any other type of acute leukemia, including
                  blastic plasmacytoid dendritic cell neoplasm

               -  Chronic myeloid leukemia with a history of blast crisis and:

                    -  With relapse or refractory disease (>= 5% marrow blasts, or circulating
                       blasts) at any time after HCT

                    -  With persistent rising minimal residual disease (defined as detectable
                       disease by morphology, flow cytometry, molecular or cytogenetic testing but
                       < 5% marrow blasts by morphology, no circulating blasts on >= 2 of two
                       consecutive tests), refractory or ineligible for treatment with tyrosine
                       kinase inhibitors at any time after HCT

               -  Myelodysplastic syndrome (MDS) of any subtype

               -  Chronic myelomonocytic leukemia (CMML)

               -  Juvenile myelomonocytic leukemia (JMML)

          -  Patients must be able to understand and be willing to give informed consent;
             decision-impaired adults may consent with their legally authorized representative;
             parent or legal representative will be asked to consent for patients younger than 18
             years old

          -  Patients must agree to participate in long-term follow-up for up to 15 years if they
             are enrolled in the study and receive T cell infusion

          -  Patients who have relapsed or have MRD after HCT may receive other agents for
             treatment of disease and remain eligible for the protocol

          -  A specific performance status score is not required for enrolling on the protocol; a
             delay in infusion of the HA-1 TCR T cells may be required for patients with low
             performance status

        DONOR SELECTION INCLUSION

          -  Donor age >= 18 years

          -  Donors must be able to give informed consent

          -  Patients must have an adult donor for HCT who is adequately HLA matched by
             institutional standards (includes HLA-matched related or unrelated donors, and
             HLA-mismatched family donors, including haploidentical donors) and is either:

               -  HLA-A*0201 positive and HA-1(H) negative (RS_1801284: G/G) or

               -  HLA-A*0201 negative

        Exclusion Criteria:

          -  Medical or psychological conditions that would make the patient unsuitable candidate
             for cell therapy at the discretion of the principal investigator (PI)

          -  Fertile patients unwilling to use contraception during and for 12 months after
             treatment

          -  Patients with a life expectancy < 3 months of enrollment from coexisting disease other
             than leukemia

          -  Patients who develop grade IV acute GVHD or severe chronic GVHD following most recent
             transplant prior to enrollment on the protocol

          -  The presence of organ toxicities will not necessarily exclude patients from enrolling
             on the protocol at the discretion of the PI; however, a delay in the infusion of HA-1
             TCR T cells may be required

        DONOR SELECTION EXCLUSION

          -  Donors who are HIV-1, HIV-2, human T-lymphotropic virus (HTLV)-1, HTLV-2 seropositive
             or with active hepatitis B or hepatitis C virus infection

          -  Unrelated donor residing outside of the United States of America (USA) unless the
             donor screening, testing and leukapheresis occur at an NMDP-affiliated and qualified
             donor center and are facilitated by the NMDP.