Pomalidomide After CAR T-cell Therapy for the Treatment of 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 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
* Co-Enrollment: Willingness to consent/ co-enroll on BMT long term follow up study, HUM00043287 (UMCC2001-0234)

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