Metabolically Fit CD19 CAR T-cell Therapy With CD34 Selection in Patients With CD19+ Relapsed/Refractory NHL, CLL/SLL

Inclusion Criteria: Patients eligible for study participation must meet all of the following criteria: 1. Disease Related Criteria Participants must have histologic confirmation of one of the following: 1. CD19+ aggressive non-Hodgkin lymphoma including any of the following subtypes * Diffuse Large B-cell Lymphoma, not otherwise specified * DLBCL, germinal-center B-cell type (GCB) * DLBCL, activated B-cell type (ABC) * T-cell histiocyte-rich B-cell lymphomas (THRBCL) * Primary cutaneous DLBCL, leg type * Intravascular large B cell lymphoma * EBV+ DLBCL, NOS * DLBCL associated with chronic inflammation * HHV8+ DLBCL, NOS * High grade B-cell lymphoma with MYC and BCL2 and/or BCL6 rearrangement(double hit lymphoma) * High grade B-cell lymphoma, NOS * Primary mediastinal B-cell lymphoma * B-cell Lymphoma, unclassifiable with features intermediate between DLBCL and Hodgkin lymphoma, as well as with features intermediate between DLBCL and Burkitt lymphoma * Follicular lymphoma grade 3B * Transformation of indolent lymphoma (i.e. CLL, MZL, FL, Waldenstrom's lymphoma,etc) to -diffuse large B-cell lymphoma * Burkitt Lymphoma * Lymphomatoid granulomatosis 2. CD19+ indolent non-Hodgkin lymphoma including any of the following subtypes: * Follicular lymphoma (grade 1-3A) * Marginal zone lymphoma: Including splenic marginal zone lymphoma, nodal marginal zone lymphoma, and mucosa associated lymphoid tissue (MALT) lymphoma * Waldenstrom's Macrogloublinemia * Nodular lymphocyte predominant hodgkin lymphoma (with documented CD19 expression) 3. Mantle cell Lymphoma 4. Chronic Lymphocytic Leukemia (CLL) or Small Lymphocytic Leukemia (SLL) 2. Prior Therapy Criteria Prior/Concurrent Therapy Related Criteria (dependent upon subtype - see below) 1. Aggressive lymphoma: patients will qualify if any of the following scenarios are met below (radiation does not count as a line of therapy) * Relapse or persistent disease after ≥ 2 lines of systemic therapy OR * Refractory disease or relapse within 12 months of completion of 1st line systemic therapy OR * Refractory disease or relapse ≥ 1 line of therapy but not a candidate for autologous stem cell transplant * Patients with Burkitt lymphoma will qualify after ≥ 1 line of therapy regardless of the timing of relapse 2. Indolent lymphoma: * Relapse or persistent disease after ≥ 2 lines of systemic therapy. (Neither single agent rituximab or radiation qualify as a line of therapy.) 3. Mantle cell lymphoma: * Relapse or persistent disease after ≥ 1 line of systemic therapy. Neither single agent rituximab or radiation qualify as a line of therapy. 4. Chronic Lymphocytic Leukemia/Small Lymphocytic Lymphoma * Evidence of progression or intolerance after ≥ 2 lines of therapy. The following will qualify as a line of therapy for CLL/SLL: * systemic chemoimmunotherapy (e.g. BR, FCR, etc), * BTK inhibitor, BCL-2 inhibitor, * or PI3 Kinase inhibitor. * Neither single agent rituximab/obinutuzumab or radiation qualify as a line of therapy. 3. Clinical/Laboratory Criteria 1. Participants must be at least 18 years old 2. Participants must have a performance status of 0-2 on the ECOG scale 3. Participants must have adequate caregiving support for CAR T-cell therapy as determined by the PI/Co-I. 4. Participants must have measurable disease on cross section imaging by PET-CT and/or CT scans alone that is at least 1.5 cm in the longest diameter and measurable in two perpendicular dimensions as defined by IWG criteria. If participants with CLL do not have measurable disease on imaging, a bone marrow biopsy showing \> 5% CLL involvement in the bone marrow will qualify for enrollment 5. Participants that have received prior CD19 targeted therapy in the past they must have a tissue biopsy after completion of CD19 targeted therapy noting CD19 expression by either flowcytometry or immunohistochemistry (IHC). CD19 expression must be sufficient per PI/Co-I 6. Adequate organ function * Bone marrow function as evidenced by the following (unless directly attributable to disease within the bone marrow) within 14 days prior to registration. * Platelet count ≥ 50,000 cells/mm3 * ANC ≥ 750 cells/mm3 * Absolute lymphocyte count ≥ 150 cells/ mm3 * Hepatic function as evidenced by the following within 14 days prior to registration. * Serum bilirubin ≤ 1.5 X ULN unless attributed to Gilbert's syndrome or hemolysis or lymphoma involvement * Cardiac * No clinically significant ECG findings per PI/Co-I * Pulmonary * Oxygen saturation \> 90% on room air * Renal function as evidenced by the following within 14 days prior to registration. * Serum creatinine ≤ 2 mg/dL or creatinine clearance (as estimated by Cockcroft Gault Equation) ≥ 60 mL/min 7. Participants with hepatitis B virus infection must have undetectable viral load and on suppressive therapy within 14 days prior to registration and no evidence of HBV related hepatic damage. 8. Participants with Hepatitis C infection must have complete eradication therapy completed, have no evidence of HCV related damage and have undetectable viral load within 14 days of registration. 9. Participants with known human immunodeficiency virus (HIV)-infection must be receiving anti- retroviral therapy and have an undetectable viral load test within 14 days prior to registration. 10. WOCBP should be advised to avoid becoming pregnant and men should be advised to not father a child while receiving treatment. All men and women of childbearing potential must use acceptable methods of birth control throughout the study as described below. FCBP must have negative serum or urine pregnancy within 7 days prior to registration. 11. Men with female partners who are of childbearing potential: Recommendations for male and partner to use at least two effective contraceptive methods, as described above, during the study. 12. Participants are able to understand and voluntarily sign consent prior to any study related assessments or procedures are performed. Exclusion Criteria: Participants eligible for study participation CANNOT meet any of the following criteria: 1. Prior/Concurrent Therapy Related Criteria Guidelines regarding when lymphoma directed therapy should be stopped prior to leukapheresis, lymphodepleting chemotherapy, and CAR T-cell infusion are detailed in the protocol. These criteria must be planned to be met for all patients. 2. Clinical/Laboratory Criteria 1. Women who are pregnant or breast-feeding. 2. Participants with active CNS lymphoma. Participants can have a history of active CNS lymphoma as outline in protocol 3. Participants with evidence of Graft vs Host Disease from allogeneic stem cell transplant are ineligible unless it is either grade 1 involvement of the skin or not requiring systemic immunosuppression. 4. Participants with uncontrolled systemic fungal, bacterial or viral infection (defined as ongoing signs/symptoms related the infection without improvement despite appropriate antibiotics, antiviral therapy and/or other treatment) 5. Participants with a history of stroke or intracranial hemorrhage within 6 months prior to registration. Any CNS disorder that would serve as a major barrier in evaluating neurotoxicity/ICANS per enrolling physician 6. Participants with prior history of malignancy other than lymphoma unless subject is free of disease for more than 1 year from signing consent. Exceptions include the following: * Basal cell carcinoma of the skin * Squamous cell carcinoma of the skin * Carcinoma in situ of the cervix or breast * Previously treated localized prostate cancer with normal PSA levels 7. Participants with primary immunodeficiency or history of autoimmune disease (e.g. Crohn's, rheumatoid arthritis, systemic lupus) requiring systemic immunosuppression/systemic disease modifying agents within the last 1 year. 8. Participants with receipt of live vaccine within 28 days prior to registration. 9. Participants with history of autologous stem cell transplant within the last 60 days or allogeneic stem cell transplant within the last 90 days or CAR T-cell therapy within the last 180 days. 10. Participants with a history of severe immediate hypersensitivity reaction to any of the agents used in the study 11. Participants with any other illness that in the opinion of the investigator, would exclude the patient from participating in this study. 12. Participants must not have evidence of active CNS lymphoma involvement. This includes parenchymal, spinal cord, meningeal, or cerebrospinal fluid involvement. Patients with history of CNS involvement must have documented remission by contrast-enhanced MRI imaging and CSF evaluation for at least 60 days prior to registration. 13. Patients must not have any unstable angina, or myocardial infarction within the last 6 months, or symptoms consistent with NYHA CHF classification III or IV 14. Participants with receipt of live vaccine within 28 days prior to registration. 15. Participants with history of autologous stem cell transplant within the last 60 days or allogeneic stem cell transplant within the last 90 days or CAR T-cell therapy within the last 180days. 16. Participants with a history of severe immediate hypersensitivity reaction to any of the agents used in the study 17. Participants with any other illness that in the opinion of the investigator, would exclude the patient from participating in this study.