Antigen-specific T Cell Therapy for AML or MDS Patients With Relapsed Disease After Allo-HCT

Inclusion Criteria: General 1. Ability to provide informed consent and documentation of informed consent prior to initiation of any study-related tests or procedures that are not part of standard-of-care for the patient's disease. Patients must also be willing and able to comply with study procedures, including the acquisition of specified research specimens 2. Age ≥ 18 years old 3. Expression of HLA-A\*0201 as determined by high resolution sequence-based typing method (e.g. TruSight HLA v2 Sequence Panel) 4. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 5. Patients must have a confirmed diagnosis of AML (according to World Health Organization (WHO) classification) or MDS with evidence of MRD or morphological relapse after HLA-matched allogeneic HCT MRD includes: 1. Detection of blasts \<5% in bone marrow 2. Detection of a clonal abnormal blast population via multi-color flow cytometry 3. Detection of known or new myeloid gene mutations 6. Donor match at 10 of 10 loci for HLA-A, -B, -C, -DRB1 and DQB1, with each typed at high resolution by DNA-based methods. a. Patients who have had prior HLA-mismatched or haploidentical allogeneic HCT will not be eligible 7. T-cell chimerism ≥ 50% to donor (by PCR analysis) 8. Acceptable laboratory parameters as follows: 1. Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) ≤ 2.5 × upper limit of normal (ULN) 2. Total bilirubin ≤ 1.5 × ULN, except patients with Gilbert's syndrome, who may enroll if the conjugated bilirubin is within normal limits 3. Creatinine \< 1.5 mg/dL or a calculated or measured creatinine clearance (CL) ≥ 50 mL/min per standard calculation 9. Recovery to Grade 1 or baseline of non-hematologic toxicities from prior treatments, excluding alopecia 10. Female patients of childbearing potential must test negative for pregnancy at enrollment and during the study. Sexually active women of child-bearing potential, unless surgically sterile, must be willing to use a highly effective method of birth control defined as those which result in a low failure rate (i.e., less than 1% per year) such as implants, injectables, combined oral contraceptives, intra-uterine devices (IUDs) or vasectomized partner 11. Male patients with partners of childbearing potential must be either vasectomized or agree to use a condom in addition to having their partners use another method of contraception resulting in a highly effective method of birth control defined as those which result in a low failure rate (i.e., less than 1% per year) such as implants, injectables, combined oral contraceptives, or IUDs. Patients should not have sexual intercourse with females who are either pregnant or lactating without double-barrier contraception Is not pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the prescreening or screening visit through one year from administration of NEXI-001 cells Exclusion Criteria: 1. Active acute or chronic graft versus host disease (GvHD) 1. Eligible patients will not be on any steroids ≥10 mg per day prednisone or equivalent or other immunosuppressants such as tacrolimus, cyclosporine, etc. 2. Intermittent topical, inhaled or intranasal corticosteroids are allowed 2. Have active or uncontrolled infections with positive cultures and/or requiring treatment with IV anti-infective agents 3. History of clinically significant cardiovascular disease including but not limited to: 1. Myocardial infarction or unstable angina within the 6 months prior to the initiation of study 2. Stroke or transient ischemic attack within 6 months prior to initiation of study 3. Clinically significant cardiac arrhythmia 4. Uncontrolled hypertension: systolic blood pressure (SBP) \> 180 mmHg, diastolic blood pressure (DBP) \> 100 mmHg 5. Congestive heart failure (New York Heart Association \[NYHA\] class III-IV) 6. Pericarditis or clinically significant pericardial effusion 7. Myocarditis 4. Clinically significant pulmonary compromise, including a requirement for supplemental oxygen use to maintain adequate oxygenation 5. History of autoimmune disease (e.g. Crohn's, rheumatoid arthritis, systemic lupus erythematosus, etc.) resulting in end organ injury or requiring systemic immunosuppression / systemic disease modifying agents within the last 2 years prior to enrollment. Subjects with a history of autoimmune-related hypothyroidism on a stable dose of thyroid replacement hormone and subjects with controlled Type 1 diabetes mellitus on a stable insulin regimen may be eligible for the study 6. Human immunodeficiency virus (HIV) seropositive; HIV testing within 2 years of enrollment 7. Seropositive for and with evidence of active viral infection with hepatitis B virus (HBV). Patients who are hepatitis B surface antigen (HBsAg) negative and HBV viral DNA negative are eligible 1. Patients who had HBV but have received an antiviral treatment and show non-detectable viral DNA for 6 months are eligible 2. Patients who are seropositive because of HBV vaccine are eligible 8. Seropositive for and with active viral infection with hepatitis C virus (HCV) a. Patients who had HCV but have received an antiviral treatment and show no detectable HCV viral DNA for 6 months are eligible 9. Second primary invasive malignancy that has not been in remission for greater than 2 years. Exceptions that do not require a 2-year remission include: resected non-melanoma skin cancer; carcinoma in situ (cervix, bladder, breast, etc.) or squamous intraepithelial lesion on Pap smear; localized prostate cancer (Gleason score \< 6); or resected melanoma in situ 10. History of trauma or major surgery within 4 weeks prior to the study enrollment 11. Any serious underlying medical or psychiatric condition that would impair the ability of the patient to receive or tolerate the planned treatment and follow up 12. Known hypersensitivity to any component of NEXI-001 T cells, cyclophosphamide, fludarabine or tocilizumab 13. Vaccination with any live virus vaccine is not permitted prior to the initiation of study treatment. Inactivated annual influenza vaccination is allowed. 14. Dementia or altered mental status that would preclude understanding and rendering of informed consent 15. History of seizures, aphasia, psychosis or other chronic clinically significant neurologic disorders a. Patients with remote history of seizures that are well controlled on anti-seizure medications and without any seizure episode for 6 months are eligible 16. Any issue that in the opinion of the investigator, would contraindicate the patient's participation in the study or confound the results of the study