A Phase 1 Study of Patient-Derived Multi-Tumor-Associated Antigen-Specific T Cells (MT-601) Administered to Patients With Relapsed or Refractory Non-Hodgkin and Hodgkin Lymphoma (APOLLO)

Inclusion Criteria: * All applicable inclusion and exclusion criteria must be met at Screening and at Baseline (re-assessment of eligibility within 14 days prior to group assignment). Participants are eligible to be included in the study only if all of the following criteria apply and the participant, in the judgement of the Investigator, is an appropriate candidate for experimental therapy: General: 1. Participant must be ≥ 18 years of age and capable of giving signed informed consent (ICF), which includes compliance with the requirements and restrictions listed in the ICF and in the protocol, at the time of signing the ICF. Disease Specific: 2. Cytologically or histologically confirmed diagnosis of NHL, HL or CLL based on the 2022 World Health Organization (WHO) criteria for hematolymphoid neoplasms 3. Enrollment of the following subtypes will be eligible: 1. LBCL including diffuse large B cell lymphoma, primary mediastinal B cell lymphoma (PMBCL), high grade B cell lymphoma (HGBL), T cell rich B cell lymphoma and transformed indolent lymphoma (transformed iNHL) 2. FL 3. MCL 4. MZL 5. HL The following additional subtypes may be enrolled in disease specific cohorts during Dose Expansion (upon approval by Sponsor) 6. CLL/SLL 7. CNS lymphoma 8. CAR T cell refractory 4. Must have measurable disease as per 2014 Lugano criteria or 2018 iwCLL criteria. Participants with splenic MZL must have measurable splenomegaly on imaging or evidence of bone marrow involvement. Prior Treatments 5. Participants who are R/R, are intolerant to, or are considered ineligible for systemic standard of care anticancer treatments, including at least 2 prior therapies. Participants who refuse standard of care treatments may also be considered if documentation is provided that he/she has been made aware of all therapeutic options. 6. For participants with LBCL, FL, and MCL: Have received CD19-directed CAR T cell therapy and relapsed ≥ 30 days or attained an incomplete response as the best response within 1 year after CAR T cell administration. Participants who refuse or are ineligible for CAR T cell therapy are eligible for this study. Note: during Dose Expansion, a specific cohort may be enrolled to evaluate participants who were refractory to CD19-directed CAR T cell therapy. Health Status 7. Karnofsky score of ≥70 or performance status of 0 or 1 on the Eastern Cooperative Oncology Group (ECOG) scale 8. Life expectancy ≥12 weeks 9. Adequate blood, liver, renal and cardiac function: 1. Hematology: Hemoglobin ≥ 7.0 g/dL (can be transfused), absolute lymphocyte count (ALC) ≥ 300/μL, (prior to apheresis only), absolute neutrophil count (ANC) ≥ 750/μL and platelet count ≥ 50,000/μL (prior to the conditioning regimen only) 2. Liver: Bilirubin ≤ 1.5X upper limit of normal (ULN) (exception of bilirubin elevation due to Gilbert's syndrome 3X); aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 3X ULN 3. Renal: Serum creatinine ≤ 1.5X ULN or measured or calculated creatinine clearance ≥ 50 mL/min (prior to the conditioning regimen) 4. Cardiac: left ventricular ejection fraction ≥ 45% (prior to the leukapheresis) Sex 10. Female: Is a woman of childbearing potential (WOCBP) and using a contraceptive method that is highly effective (i.e., with a failure rate of \< 1% per year), preferably with low user dependency during the intervention period and for at least 6 months after the last infusion of MT-601 and agrees not to donate eggs (i.e., ova and oocytes) for the purpose of reproduction during this period 11. Male participants are eligible to participate if they agree to the following during the intervention period and for at least 6 months after the last infusion of MT-601: Refrain from donating sperm PLUS either: Be abstinent from intercourse as their preferred and usual lifestyle (abstinent on a long-term and persistent basis) and agree to remain abstinent OR Must agree to use a male condom AND should also be advised of the benefit for a nonpregnant female partner to use a highly effective method of contraception as a condom may break or leak Exclusion Criteria: * Patients are excluded from the study if any of the following criteria apply: Disease-related 1. Evidence of bulky disease at the time of the conditioning regimen (≥ 10 cm in diameter for LBCL or HL and \> 6 cm for other subtypes) 2. Untreated or ongoing treatment for CNS lymphoma or completed treatment within 2 weeks of apheresis (Note: May be allowed in Dose Expansion if disease specific cohort for CNS lymphoma is opened) 3. Refractory to CAR T therapy defined as a best response of stable disease or disease progression (Note: May be allowed in Dose Expansion if disease specific cohort for CAR T cell therapy refractory is opened) 4. Requirement for urgent therapy due to tumor mass effects such as bowel obstruction or blood vessel compression Medical Conditions 5. Primary immunodeficiency 6. Severe or uncontrolled autoimmune disorder 7. History or presence of clinically relevant CNS pathology such as epilepsy, seizure, aphasia, stroke, severe brain injuries, dementia, Parkinson's disease, cerebellar disease, organic brain syndrome, or psychosis 8. Unresolved immune effector cell-associated neurotoxicity syndrome (ICANS) from prior CAR T cell administration. Consideration for Grade 1 may be made after discussion with the Medical Monitor 9. History of malignancy other than nonmelanoma skin cancer or carcinoma in situ (e.g., cervix, bladder, breast, and/or prostate) unless disease free for at least 3 years 10. Cardiac conditions: 1. Medically uncontrolled hypertension (≥ 160 mmHg systolic blood pressure or ≥ 100 mmHg diastolic blood pressure) 2. Congestive heart failure Class ≥ II as defined by the New York Heart Association 3. Acute coronary syndrome (including unstable angina, coronary artery stenting, or angioplasty, bypass grafting within prior 6 months) 4. History or evidence of current, uncontrolled, clinically significant, unstable arrhythmias 11. Oxygen saturation at room air \< 92% 12. Participant has known human immunodeficiency virus (HIV) infection, or active hepatitis B virus (HBV)/hepatitis C virus (HCV) infection 13. Acute bacterial, viral, fungal infection requiring systemic therapy (uncomplicated urinary tract infection and bacterial pharyngitis are permitted if responding to therapy) 14. History of severe allergic reactions to any of the study intervention components including conditioning regimen, dimethyl sulfoxide (DMSO) or to tocilizumab 15. Clinically significant reversible toxicities from prior cancer therapy that have not recovered to Grade 1 or baseline * Participants with Grade 2 neuropathies due to prior treatment will be allowed on study. * Participants with clinical nonsignificant toxicities, such as alopecia, will be allowed on study. Prior/Concomitant Therapy Prior to Apheresis: 16. Receipt of allogeneic hematopoietic cell transplant (HCT) within 12 months; on immunosuppression or with evidence of donor/mixed chimera 17. Receipt of autologous HCT within 3 months 18. Treatment with CD19-directed CAR T cell therapy within 3 months 19. Treatment with bispecific antibody within 1 month 20. Treatment with antibody drug conjugates (ADC's) or PD-1/PD-L1 within 21 days 21. Treatment with monoclonal antibodies impacting T cell function within 14 days 22. Treatment with systemic immunosuppression including systemic corticosteroids (unless ≤5 mg/day oral prednisone or steroid equivalent) within 14 days 23. Treatment with chemotherapy within 7 days Prior to the conditioning regimen: 24. Treatment with a live, attenuated vaccine within 4 weeks 25. Treatment with antibody drug conjugates (ADC's) or PD-1/PD-L1 within 21 days 26. Treatment with chemotherapy or biologics/monoclonal antibodies within 14 days 27. Treatment with radiation therapy within 7 days 28. Treatment with a tyrosine kinase inhibitor (TKI) within 7 days or 5 half-lives (whichever is longer) before conditioning regimen 29. Hematopoietic growth factors \<2 days At either time: 30. Treatment with experimental CAR T cell product unless approved by Medical Monitor 31. Treatment with other cancer therapy including investigational agents that do not fit in the above categories within 14 days 32. Major surgery within 14 days Other 33. Pregnant or lactating 34. Any other issue which, in the opinion of the treating physician, would make the participant ineligible for the study