Phase I Study of Adoptive Immunotherapy With CD8+ and CD4+ Memory T Cells Transduced to Express an HA-1-Specific T Cell Receptor (TCR) for Children and Adults With Recurrent Acute Leukemia After Allogeneic Hematopoietic Stem Cell Transplantation (HCT)

Inclusion Criteria: * Subject age 0-80 years at the time of enrollment. * Subject must express HLA-A\*0201 * Subject must have the HA-1(H) genotype (RS\_1801284: A/G, A/A) * Subject 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 * Subjects 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) * Subjects 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 subjects younger than 18 years old * Subjects 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 * Subjects 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 subjects with low performance status DONOR SELECTION INCLUSION * Donor age \>= 18 years * Donors must be able to give informed consent Exclusion Criteria: * Medical or psychological conditions that would make the subject unsuitable candidate for cell therapy at the discretion of the principal investigator (PI) * Fertile subjects unwilling to use contraception during and for 12 months after treatment * Subjects with a life expectancy of \< 3 months of enrollment from coexisting disease other than leukemia * Subjects who have ongoing grade IV acute GVHD or severe chronic GVHD following most recent transplant. Exception: the principal investigator (PI) may make an exception on a case-by-case basis to include such a subject if there is doubt surrounding the GVHD diagnosis and/or sustained significant improvement in GVHD severity * The presence of organ toxicities will not necessarily exclude subjects 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 human immunodeficiency virus (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 National Marrow Donor Program (NMDP)-affiliated and qualified donor center and are facilitated by the NMDP