This phase I trial tests the safety and effectiveness of pomalidomide after CD19 chimeric antigen receptor T-cell (CD19CART) therapy for the treatment of patients with CD19+ B-cell leukemias or lymphomas that have come back after a period of improvement (relapsed) or do not respond to treatment (refractory). Chimeric antigen receptor (CAR) T-cell therapy is a type of treatment in which a patient's T-cells (a type of immune system cell) are changed in the laboratory so they will attack cancer cells and are then re-infused into the patient. Following CAR T-cell infusion, CAR T-cells must expand and persist in the blood stream in order to most effectively treat leukemia/lymphoma. Pomalidomide stops the growth of blood vessels, stimulates the immune system, and may kill cancer cells. Research has shown that drugs like pomalidomide can modify the immune system and increase the number or improve the function of CAR T-cells in the blood. Pomalidomide may enhance the treatment effects of CAR T-cell therapy in patients who have received CD19CART therapy for relapsed or refractory CD19+ B-cell leukemia or lymphoma.

Inclusion Criteria:

• Subject must have had a histologically or cytologically confirmed R/R CD19+ B-cell leukemia or lymphoma and have received a commercially available CAR-T product approved to treat R/R CD19+ Bcell leukemias and lymphomas.
• Subject must be 28 - 56 days post infusion of CD19CART product at time of enrollment.
• >= 18 years in age at time of enrollment
• Subject is able to swallow pills/tablets
• Karnofsky or Lansky performance score of >= 50%
• Absolute neutrophil count (ANC) >= 750/mm^3 (granulocyte colony stimulating factor allowed)
• Platelets >= 50,000/mm^3 (transfusion independent for >= 7 days, defined as not receiving platelet transfusions for at least 7 days prior to enrollment, unless due to marrow involvement from primary malignancy [thrombopoietin (TPO) mimetics allowed])
• Total bilirubin =< 1.5 x upper limit of normal (ULN) per institution
• Alanine aminotransferase (ALT [serum glutamate pyruvate transaminase (SGPT)]) =< 3 x institutional ULN per institution
• Serum albumin >= 2.0 g/dL
• Creatinine clearance (Cockcroft-Gault equation) >= 30 mL/min/1.73 m^2
• Sexually active females capable of becoming pregnant and males must agree to participate in the pomalidomide Risk Evaluation and Mitigation Strategy (REMS) program
• Patients must agree not to donate blood during treatment with pomalidomide and for 4 weeks following discontinuation of the drug because the blood might be given to a pregnant female patient whose fetus must not be exposed to pomalidomide

Exclusion Criteria:

• Patients with known progressive or refractory disease.

• The following transplant or CAR T-related events are excluded:

  • Active grade >= 2 acute or chronic graft versus host disease (GVHD)
  • Active cytokine release syndrome (CRS) grade >= 2
  • Active immune effector cell associated neurotoxicity (ICANS) grade >= 2

• Subject receiving >= 0.25 mg/kg/day of methylprednisolone equivalent. Subject being treated with medications with a known major drug interaction to pomalidomide. Specifically, patients receiving CYP1A2 inhibitors, such as ciprofloxacin, omeprazole, cimetidine, estrogen, and fluvoxamine.

• Patient who smokes cigarettes.

• Subject must not have initiated or received intervening therapy for a primary or secondary malignancy within 28 days of study enrollment, including, a) myelosuppressive chemotherapy, b) biologic anti-neoplastic agents (e.g., ruxolitinib, imatinib, dasatinib…), or checkpoint inhibitors (e.g., pembrolizumab). The use of cytokine inhibition for management of CRS/ICANS is allowed within the prior 28 days

• Receipt of radiation therapy (XRT) (focal or large field, including cranial or cranial-spinal) within 28 days prior to enrollment

• Stem cell transplant or rescue following most recent CD19CART therapy

• History of allergic reactions to pomalidomide or any of the excipients and any similar compounds

• Intercurrent illness or conditions:

  • Patients with uncontrolled infections. In addition, patients with any documented bacteremia, fungemia, or new onset viremia that requires antimicrobial therapy within 72 hours prior to enrollment. Empiric antimicrobials are allowed
  • Active grade >= 4 gastrointestinal, hepatic, pulmonary, renal, cardiac toxicity by Common Terminology Criteria for Adverse Events (CTCAE) version (v)5.0 criteria. Patients requiring dialysis are excluded
  • Patients with concomitant genetic syndromes associated with bone marrow failure states: such as patients with Fanconi anemia, severe congenital neutropenia, Schwachman syndrome or any other known bone marrow failure syndrome. Patients with Down syndrome are not excluded
  • History of known prior arterial thromboembolism, venous thromboembolism, pulmonary embolism, cardiovascular accidents, or myocardial infarctions within 3 months prior to enrollment

• Pregnant women are excluded from this study. Women should discontinue breastfeeding during treatment and for at least 4 weeks after discontinuation of study drug

• HIV positivity within 8 weeks of screening on polymerase chain reaction (PCR) based assay

Individual subject data will not be provided; however, our cumulative findings based upon response and survival for the entire population will be provided.