Provision of TCRγδ T Cells and Memory T Cells Plus Selected Use of Blinatumomab in Naïve T-cell Depleted Haploidentical Donor Hematopoietic Cell Transplantation for Hematologic Malignancies Relapsed or Refractory Despite Prior Transplantation

Inclusion Criteria for Transplant Recipient:

* Age less than or equal to 21 years.
* Any of the following hematologic malignancies that has relapsed or remains refractory after prior allogeneic HCT (this includes any stage of disease - such as refractory due to induction failure, refractory in relapse, or in any CR - as long as the hematologic malignancy remained persistent or returned after a previous allogeneic HCT):

  * ALL, AML, Myeloid Sarcoma, CML, Juvenile myelomonocytic leukemia (JMML), myelodysplastic syndrome (MDS), non-Hodgkin lymphoma (NHL)
* Has a suitable single haplotype matched (≥ 3 of 6) family member donor.
* Does not have any other active malignancy other than the one for which this transplant is indicated.
* If prior CNS leukemia, it must be treated and in CNS CR
* Does not have current uncontrolled bacterial, fungal, or viral infection.
* There is no minimum time from the previous transplant, but patients must meet the following criteria:

  * Left ventricular ejection fraction \> 40%, or shortening fraction ≥ 25%.
  * Creatinine clearance (CrCl) or glomerular filtration rate (GFR) ≥ 50 ml/min/1.73m2.
  * Forced vital capacity (FVC) ≥ 40% of predicted value; or pulse oximetry ≥ 92% on room air if patient is unable to perform pulmonary function testing.
  * Karnofsky or Lansky (age-dependent) performance score ≥ 50 (See Appendix A).
  * Bilirubin ≤ 3 times the upper limit of normal for age.
  * Alanine aminotransferase (ALT) or aspartate aminotransferase (AST) ≤ 5 times the upper limit of normal for age.
  * Not pregnant. If female with child bearing potential, must be confirmed by negative serum or urine pregnancy test within 14 days prior to enrollment.
  * Not breast feeding

Inclusion Criteria for Haploidentical Donor:

* At least single haplotype matched (≥ 3 of 6) family member
* At least 18 years of age.
* HIV negative.
* Not pregnant as confirmed by negative serum or urine pregnancy test within 14 days prior to enrollment (if female).
* Not breast feeding.
* Regarding donation eligibility, is identified as either:

  * Completed the process of donor eligibility determination as outlined in 21 CFR 1271 and agency guidance; OR
  * Does not meet 21 CFR 1271 eligibility requirements, but has a declaration of urgent medical need completed by the principal investigator or physician sub-investigator per 21 CFR 1271.