A patient cannot be considered eligible for this study unless ALL of the following conditions are met. ° Disease type Cohort 1, High Risk Disease: Patients with age ≤ 26 years at the time of informed consent with no available and suitably matched related or unrelated donor within 4 weeks, with one of the following diagnoses: I. Acute myelogenous leukemia (AML): - Complete first remission (CR1) with blast count < 5% by bone marrow morphology at high risk for relapse such as any of the following: - Known prior diagnosis of myelodysplasia (MDS) - High risk cytogenetics (e.g., those associated with MDS, abnormalities of 5, 7, 8, complex karyotype) and/or high-risk molecular abnormalities (e.g., TP53) - Requirement for 2 or more inductions to achieve CR1 - Therapy-related AML (t-AML) or therapy-related myeloid neoplasm (t-MN) (including after therapy for other malignancy, and/or gene therapy or cell therapy) - Presence of Minimal/Measurable Residual Disease (MRD+) by cytogenetics, flow cytometry or molecular methods (at End of Induction or End of Consolidation) - Other high-risk features not defined above. - Complete second remission (CR2) or subsequent remission, with blast count < 5% by bone marrow morphology - Presence of MRD by multiparameter flow cytometry at pre-transplant evaluation is acceptable. II. Acute lymphoblastic leukemia (ALL): - Complete first remission (CR1) with MRD negative status by multicolor flow cytometry, at high risk for relapse such as any of the following: - Presence of any high risk cytogenetic abnormalities such as t(9;22), t(1;19), t(4;11) or other, KMT2A (11q23) or other high risk molecular abnormality - Failure to achieve complete remission (CR) after four weeks of induction therapy (transplant to follow antibody therapy and/or CAR T cells) - Persistence or recurrence of MRD on therapy (Transplant to follow antibody therapy and/or CAR T cells) - T-ALL in CR even with presence of MRD - Other high-risk features not defined above - Complete second remission (CR2) or subsequent remission with MRD negative status by multiparameter flow cytometry. - Relapse in less than 36 months from CR1 - Relapse for T-ALL - Patients after antibody therapy (e.g., blinatumomab, inotuzumab, other) and/or CAR-T cell therapy that resulted in MRD negative status by multiparameter flow cytometry. III. Other acute leukemias: - Leukemias of ambiguous lineage or of other types with < 5% blasts by bone marrow morphology. - Patients with persistent/relapsed disease with cytogenetic, flow cytometric or molecular aberrations in < 5% of cells. - Chronic myelogenous leukemia: Patients with history of blast crisis or accelerated phase. - Any leukemia that developed after gene therapy or cell therapy IV. Myelodysplastic Syndrome (MDS): - Any IPSS risk category with life-threatening cytopenia(s). - Any IPSS risk category with high risk cytogenetic/molecular findings (5, 7, 8, complex karyotype, or TP53) V. Non-Hodgkin lymphoma (NHL) or Hodgkin lymphoma (HL) at high risk of relapse or progression if not in remission: - Patients with aggressive histology (such as, but not limited to, diffuse large B-cell NHL, mantle cell NHL, and T-cell NHL) in CR. - Patients with indolent B cell NHL (such as, but not limited to, follicular, small cell or marginal zone NHL) will have 2nd or subsequent progression with stable disease/ CR/ PR with no single lesion equal to or more than 5 cm. - Patients with HL without progression of disease (POD) after salvage chemotherapy with no single lesion ≥5 cm. Cohort 2: Very High-Risk disease: 1. Patients in CR (bone marrow blasts <5% by morphology) who had prior allogeneic transplant and disease recurrence. The second transplant will take place at least 4 months after the first. - Acute myelogenous leukemia (AML) or Myelodysplastic Syndrome (MDS): Relapse after previous transplant, in CR after induction therapy. MRD positive status by multi-parameter flow cytometry is accepted. - Acute lymphoblastic leukemia (ALL): Relapse after previous transplant, in CR after induction therapy and/or antibody therapy/CAR T cells. MRD positive status after targeted therapy, as evaluated by multi-parameter flow cytometry is accepted. - Other: patients with leukemia or lymphoma, who, in the opinion of their physician, are not likely to have reduction in disease burden with further chemotherapy. 2. Patients with relapsed/refractory disease at either first or second allogeneic transplant, with up to 30% bone marrow blasts by multiparameter flow cytometry or morphology. ° Relapse after previous transplant with < 30% blasts by bone marrow morphology, or with cytogenetic, flow cytometric, or molecular abnormalities in < 30% of bone marrow cells, after induction therapy. ° Primary refractory or relapsed AML with < 30% blasts by bone marrow morphology or with cytogenetic, flow cytometric, or molecular abnormalities in < 30% of bone marrow cells. ° Age 0-26 years at the time of informed consent ° Performance: Karnofsky (≥16 years) or Lansky score (<16 years) of ≥70% (see Appendix A). ° Not Pregnant and Not Nursing ° Required Organ Function - Bilirubin ≤ 1.5 mg/dL (unless benign congenital hyperbilirubinemia). - ALT ≤ 3 x upper limit of normal. - Pulmonary function (FVC, FEV1 and DLCO corrected for hemoglobin) ≥ 50% predicted. - In young children unable to perform pulmonary function testing: pulse oximetry >92% in room air, and a normal CT of the chest (if CT is not normal, the child needs to be evaluated and cleared by pediatric pulmonary physician). - Left ventricular ejection fraction > 50%. - Age-adjusted Hematopoietic Cell Transplantation-Comorbidity Index (aaHCT-CI) ≤ 7. - Female patients of childbearing potential must have a negative serum pregnancy test within 7 days of enrolment and must be willing to use an effective contraceptive method while enrolled in the study. - Renal: Serum creatinine (SCr) ≤ 1.5 x normal for age. If SCr is outside the normal range, then CrCl > 50 mL/min (calculated or estimated) or estimated GFR (mL/min/1.73m2) >30% of predicted normal for age. Normal GFR by Age : Mean GFR +- SD (mL/min/1.73m^2) 1 week : 40.6 + / - 14.8 2-8 weeks : 65.8 + / - 24.8 >8 weeks : 95.7 + / - 21.7 2-12 years : 133.0 + / - 27.0 13-21 years (males) : 140.0 + / - 30.0 13-21 years (females) : 126.0 + / - 22.0 GFR, glomerular filtration rate; SD, standard deviation; Greater than 2 years old: Normal GFR is 100 mL/ min; Infants: GFR must be corrected for body surface area.

Exclusion criteria for both cohorts: ° Inadequate performance status/ organ function. ° Active CNS leukemic involvement. - Chloroma >2 cm. - Active and uncontrolled infection (bacterial/fungal/viral) at time of transplant. - HIV infection. - Seropositivity for HTLV-1. - Pregnancy or breast feeding. - Patient or guardian unable to give informed consent or unable to comply with the treatment protocol including appropriate supportive care, long-term follow-up, and research tests. - Any abnormal condition or lab result that is considered by the PI capable or altering patient's condition or study outcome. Cohort 2 Very High-Risk Disease (additional to above): ° Allogeneic HCT in the preceding 4 months. Note (1): Prior checkpoint inhibitors/blockade in the last 12 months: eligibility to be discussed with study PI. Note (2): For patients with known HBV and/or HCV infection : - For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. - Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load.