Belinostat in Combination With Azacitidine or Pralatrexate for the Treatment of Relapse or Refractory T-cell Lymphoma

Inclusion Criteria: * Ability to understand and willingness to sign a written informed consent document * Documented informed consent of the participant and/or legally authorized representative. Assent, when appropriate, will be obtained per institutional guidelines * Age: ≥ 18 years * Have a performance status of 0-1 on the Eastern Cooperative Oncology Group (ECOG) Scale (performance status \[PS\]) at time of enrollment. Patients with a performance status of 2 on the ECOG Scale due to lymphoma may be eligible with principal investigator (PI) approval * Histologically confirmed PTCL in the following subtypes below (by local review). Eligible histologies include: * Arm A * Histologically confirmed TFH cell lymphomas. Nodal TFH cell lymphomas encompasses three subtypes: * Angioimmunoblastic T-cell lymphoma (AITL)(World Health Organization \[WHO\]4R)/follicular helper T-cell lymphoma (TFH lymphoma), angioimmunoblastic type (ICC)/nodal TFH cell lymphoma, angioimmunoblastic-type (WHO5) * Nodal PTCL with TFH phenotype (nodal PTCL, TFH)(WHO4R)/TFH lymphoma, NOS (ICC)/nodal TFH cell lymphoma, not otherwise specified (NOS) (WHO5) * Follicular T-cell lymphoma (FTCL)(WHO4R)/TFH lymphoma, follicular type (ICC)/ nodal TFH cell lymphoma, follicular-type (WHO5) * Arm B * Peripheral T-cell lymphoma not otherwise specified (PTCL-NOS) * Monomorphic epitheliotropic intestinal T-cell lymphoma (MEITL) * Enteropathy-associated T-cell lymphoma (EATL) * Anaplastic large cell lymphoma (ALCL) * Hepatosplenic T-cell lymphoma * Cutaneous T-cell lymphoma (CTCL) with large cell transformation * Subcutaneous panniculitis-like T-cell lymphoma * Primary cutaneous CD8+ aggressive epidermotropic cytotoxic T-cell lymphoma (CD8+ PCAETL) * Primary cutaneous gamma-delta T-cell lymphoma (PCGDTL) * Must have received at least one prior systemic therapy and have an indication for treatment * NOTE: For systemic ALCL, prior systemic therapy must have included brentuximab vedotin * Measurable disease, including at least 1 nodal site measuring ≥ 1.5cm or 1 extranodal site measuring 1.0 cm in longest dimension on CT or FDG-PET or marrow-only disease (disease only found on bone marrow biopsy) * Life expectancy \> 12 weeks * Without bone marrow involvement: Absolute neutrophil count (ANC) ≥ 1,000/mm\^3 * NOTE: Growth factor is not permitted within 7 days of ANC assessment unless cytopenia is secondary to disease involvement * With bone marrow involvement: ANC ≥ 750/mm\^3 * NOTE: Growth factor is not permitted within 7 days of ANC assessment unless cytopenia is secondary to disease involvement * Without bone marrow involvement: Platelets ≥ 75,000/mm\^3 * NOTE: Platelet transfusions are not permitted within 7 days of platelet assessment unless cytopenia is secondary to disease involvement * With bone marrow involvement: Platelets ≥ 50,000/mm\^3 * NOTE: Platelet transfusions are not permitted within 7 days of platelet assessment unless cytopenia is secondary to disease involvement * Hemoglobin ≥ 8g/dL * NOTE: Red blood cell transfusions are not permitted within 7 days of hemoglobin assessment unless cytopenia is secondary to disease involvement * Total bilirubin ≤ 1.5 x upper limit of normal (ULN) * NOTE: Patients with documented Gilbert's disease, documented liver or pancreatic involvement in lymphoma may be enrolled if total bilirubin ≤ 3.0 x ULN OR direct bilirubin ≤ ULN for patients with total bilirubin levels \> 1.5 ULN * Aspartate aminotransferase (AST) ≤ 3.0 x ULN OR ≤ 5 x ULN for patients with liver involvement by lymphoma * Alanine aminotransferase (ALT) ≤ 3.0 x ULN OR ≤ 5 x ULN for patients with liver involvement by lymphoma * Measured or calculated creatinine clearance ≥ 60 mL/min (glomerular filtration rate \[GFR\] can also be used in place of creatinine clearance \[CrCl\]) * If not receiving anticoagulants: International normalized ratio (INR) OR prothrombin (PT) ≤ 1.5 x ULN * If on anticoagulant therapy: PT must be within therapeutic range of intended use of anticoagulants * If not receiving anticoagulants: Activated partial thromboplastin time (aPTT) ≤ 1.5 x ULN * If on anticoagulant therapy: aPTT must be within therapeutic range of intended use of anticoagulants * Women of childbearing potential (WOCBP): Negative urine or serum pregnancy test * If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Agreement by females and males of childbearing potential to use an effective method of birth control or abstain from heterosexual intercourse for the course of the study and after completion of study treatment as described below separately for males and females * WOCBP must remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods that result in a failure rate of ≤ 1% per year during the treatment period and for at least 1 month after the last dose of study treatment. Women must refrain from donating eggs during this same period * Examples of contraceptive methods with a failure rate of ≤ 1% per year include bilateral tubal ligation, male sterilization, hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices * The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient * Periodic abstinence (e.g., calendar, ovulation, symptothermal, or post-ovulation methods) and withdrawal are not acceptable methods of contraception * For male participants: agreement to remain abstinent (refrain from heterosexual intercourse) or use a condom, and agreement to refrain from donating sperm, as defined below: * With female partners of childbearing potential or pregnant female partners, men must remain abstinent or use a condom during the treatment period and for at least 1 month after the last treatment. Men must refrain from donating sperm during this same period * The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not acceptable methods of preventing drug exposure * Childbearing potential defined as being post-menarcheal (women only), not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) * Diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) within 7 days prior to cycle 1 day 1, for indications other than lymphoma symptom control. Patients who require lymphoma symptom control during screening may receive steroids in the following manner: * Up to 30 mg/day of prednisone or equivalent may be used for lymphoma symptom control during screening, including prior to finalization of staging (not included as part of pre-phase treatment). If glucocorticoid treatment is urgently required at higher doses for lymphoma symptom control prior to the start of study treatment, tumor assessments must be completed prior to initiation of \> 30-100 mg/day of prednisone or equivalent. Prednisone \> 30-100 mg/day or equivalent may be given for a maximum of 7 days as a pre-phase treatment. If patients exceed the allowed dosing of corticosteroids, patients may still be eligible for the study provided that baseline imaging is performed (or repeated) after completion of the course of higher dose of steroids. Allowed corticosteroid dosing resets from the point of imaging forward and patients who do not exceed the allowed corticosteroid dosing from the point of imaging until initiation of study treatment may enroll. Steroids at a dose less than the equivalent of 10 mg/day of prednisone and inhaled, nasal, and topical steroids are permitted. Adrenal replacement steroid doses \> 10 mg daily prednisone equivalent in the absence of active autoimmune disease are permitted. Treatment with a short course of steroids (\< 7 days) up to 7 days prior to cycle 1 day 1 is permitted Exclusion Criteria: * Patients who received prior therapy with belinostat or azacitidine (Arm A) or belinostat or pralatrexate (Arm B) without having had evidence of objective response (i.e. patients whose best response was stable disease or progressive disease) * Note: Patients who previously responded to any of these agents are eligible. Their last dose of either belinostat, azacitidine or pralatrexate must be \> 14 days prior to day 1 of protocol therapy * Chemotherapy, radiation therapy, biological therapy, immunotherapy within 14 days or five half-lives (whichever is shorter for non-radiation therapy) prior to day 1 of protocol therapy * Prior autologous stem cell transplantation within 60 days of day 1 of protocol therapy * Major surgery within 4 weeks prior to the start of cycle 1, other than for diagnosis confirmation * UGT1A1 inhibitors within 14 days prior to day 1 of protocol therapy * History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agents * Known active bacterial, viral, fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds) which requires systemic treatment. Patients may proceed with screening during treatment for infection, but systemic treatment must be completed by cycle 1 day 1 * If a patient has signs/symptoms suggestive of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection, the patient must have a negative molecular (e.g., polymerase chain reaction \[PCR\]) test or 2 negative antigen test results at least 24 hours apart. Patients who do not meet SARS-CoV-2 infection eligibility criteria must be screen-failed and may only be re-screened if the following have been met: * At least 10 days since first positive test result have passed in asymptomatic patients or at least 10 days since recovery, defined as resolution of fever without use of antipyretics and improvement in symptoms * Subjects with concurrent active hepatitis B (defined as hepatitis B virus surface antigen \[HBsAg\] positive and/or detectable hepatitis B virus \[HBV\] DNA) and/or hepatitis C virus (defined as anti-hepatitis C virus \[HCV\] antibody \[Ab\] positive and detectable HCV ribonucleic acid \[RNA\]) infection. * Hepatitis B and C screening tests are not required unless: (1) Known history of HBV and HCV infection or (2) As mandated by local health authority * Subjects with HIV infection * Other active malignancy. Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * Note: If such malignancies were treated with either belinostat, azacitidine, or pralatrexate the 14 day washout applies * Females only: Pregnant or breastfeeding * Known active central nervous system lymphoma * Participants who are receiving other investigational agents * Any other condition that would, in the investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)