Co-Transplant of an Unmodified Haplo-Identical Graft With Cord Blood

The purpose of this study is to see if see if adding the specific combination of donors can result in acceptable levels of survival without evidence of disease.

Inclusion Criteria: * Participants with the following hematologic malignancies: * Acute myelogenous leukemia (AML): High-risk AML including: * Antecedent hematological disease (e.g., myelodysplasia (MDS)) * Treatment-related * Complete Remission (CR1) with poor or intermediate-risk cytogenetics or molecular markers (e.g. Flt 3 mutation, 11q23, del 5, del 7, TP53 mutations, complex cytogenetics) * Participants must be in CR1, CR2, CR3 or CRi * Acute lymphoblastic leukemia (ALL) * High-risk CR1 including: * Poor-risk cytogenetics (e.g., t(9;22)or 11q23 rearrangements) * Presence of minimal disease by flow cytometry or PCR or Clonoseq after 2 or more cycles of chemotherapy * No CR within 4 weeks of initial treatment * Participants in CR2 or beyond * Participants must be in CR1, CR2, CR3, or CRi * Myelodysplastic syndromes (MDS), Intermediate, High or Very High Risk by the revised international prognostic scoring system (IPSS-R) or treatment related MDS * High-risk lymphoma * Age \> 18 years * Participants without a suitable HLA-matched related or unrelated donor CASE9Z24 Page 17 Version dated 12.16.2025 * Participants with the following suitable grafts: * A 4-8/8 HLA high resolution matched cord blood unit with a cell dose of 1.0x105 CD34 cells/kg. * A haplo-identical donor with a goal cell dose of \> 4.0x106 CD34cells/kg (minimum 2 x106 CD34 cells/kg) * Concurrent Therapy for Extramedullary Leukemia or CNS Lymphoma: Concurrent therapy or prophylaxis for testicular leukemia, CNS leukemia including standard intrathecal chemotherapy and/or radiation therapy will be allowed as clinically indicated. Such treatment may continue until the planned course is completed. Participants must be in CNS remission at the time of protocol enrollment if there is a history of CNS involvement. Maintenance therapy after transplant is allowed. * Participants must have the ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Participants with inadequate Organ Function as defined by: * Creatinine clearance \< 40ml/min (Cockcroft-Gault) * Bilirubin \> 2X institutional upper limit of normal unless Gilbert syndrome * AST (SGOT) \> 3X institutional upper limit of normal * ALT (SGPT) \> 3X institutional upper limit of normal * Pulmonary function: DLCOc \< 60% * Cardiac: left ventricular ejection fraction \< 40% * ECOG \