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 tr