Pembrolizumab and Mogamulizumab in Advanced-stage, Relapsed/Refractory Cutaneous T-cell Lymphomas

Inclusion Criteria: * Written informed consent and HIPAA authorization for release of personal health information prior to registration. NOTE: HIPAA authorization may be included in the informed consent or obtained separately. * Age ≥ 18 years at the time of consent. * ECOG Performance Status of ≤ 1 within 7 days prior to Cycle 1 Day 1 treatment. * Histological confirmation of cutaneous T-cell lymphoma (Mycosis Fungoides/Sezary Syndrome) with Stage IIB-IVB disease (TNMB Classification). * Measurable disease according to Modified Severity Weighted Assessment Tool (mSWAT) within 30 days prior to treatment. * Patients must have measurable, unirradiated disease. Prior disease radiation, if greater than 7 days prior to C1D1, is acceptable (see protocol). However, patient must have measurable disease that has not been radiated. * Patients must have failed at least one prior line of systemic therapy. This includes ECP. Prior cancer treatment must be completed at least 28 days prior to Cycle 1 Day 1(C1D1) and the subject must have recovered from all reversible acute toxic effects of the regimen (other than alopecia) to ≤ grade 1 or baseline. * Archival tissue is required and will be identified at screening and shipped prior to C1D1 (10-15 unstained slides; obtained within 90 days of registration). Subjects that do not have archival tissue will be required to undergo a skin biopsy. * Systemic 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 (\< 5 days) up to 7 days prior to study registration is permitted. * Demonstrate adequate organ function as defined below. All screening labs to be obtained within 28 days prior to Cycle 1 Day 1. * Hematological * Absolute Neutrophil Count (ANC) ≥ 500/µL * Hemoglobin (Hgb) ≥ 8 g/dL * Platelet Count ≥ 25 000/µL * Renal ---Creatinine OR Measured or calculated creatinine clearance1 ≤ 1.5 × ULN OR ≥ 30 mL/min for participant with creatinine levels \> 1.5 × institutional ULN * Hepatic * Bilirubin ≤ 1.5 ×ULN OR direct bilirubin ≤ ULN for participants with total bilirubin levels \> 1.5 × ULN * Aspartate aminotransferase (AST) ≤ 2.5 × ULN (≤ 5 × ULN for participants with liver metastases) * Alanine aminotransferase (ALT) ≤ 2.5 × ULN (≤ 5 × ULN for participants with liver metastases) * Coagulation ---International Normalized Ratio (INR) or Prothrombin Time (PT) Activated Partial Thromboplastin Time (aPTT) ≤1.5 × ULN unless participant is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants * Females of childbearing potential must have a negative serum pregnancy test within 72 hours prior to registration. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. See protocol for definition of childbearing potential. * Females of childbearing potential must be willing to abstain from heterosexual intercourse or to use an effective method(s) of contraception as outlined in protocol. Males must be willing to abstain from heterosexual intercourse or to use an effective method(s) of contraception as outlined in protocol. * Subjects with CNS disease are eligible so long as they meet all other eligibility criteria. * Patients must have a life expectancy of at least 6 months. * As determined by the enrolling physician or protocol designee, ability of the subject to understand and comply with study procedures for the entire length of the study. Exclusion Criteria: * Patients with a known history of Human Immunodeficiency Virus (HIV) infection are excluded. NOTE: No HIV testing is required unless mandated by local health authority. * Patients with concurrent active Hepatitis B (defined as HBsAg positive and/or detectable HBV DNA) and Hepatitis C virus (defined as anti-HCV Ab positive and detectable HCV RNA) infection. NOTE: 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. * Patients previously treated with checkpoint blockade, including pembrolizumab, or mogamulizumab, are excluded. * Patients who have received disease radiation therapy within 7 days of C1D1. * Has a diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to Cycle 1 Day 1. Systemic steroids at a dose less than the equivalent of 10 mg/day of prednisone and inhaled, nasal, and topical steroids are permitted as detailed in protocol. * Has active or prior autoimmune disease or inflammatory disorders that have required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs) with teh exception of vitiligo or alopecia. Replacement therapy (eg., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment and is allowed. * Has a history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease. * Known additional malignancy that is progressing or has required active treatment within the past 2 years. NOTE: patients with non-melanoma skin cancers and in situ cancers that do not require systemic therapies are eligible. * Active infection requiring systemic therapy. * Prior allogeneic stem cell transplant or allogeneic cellular therapies, recent immunosuppressive therapies (for any reason). * Prior solid organ transplant. * Pregnant or breastfeeding (NOTE: breast milk cannot be stored for future use while the mother is being treated on study). * Has severe hypersensitivity (≥Grade 3) to pembrolizumab or mogamulizumab and/or any of their excipients. * Treatment with any investigational drug or investigation device within 30 days prior to registration. * Has received a live vaccine or live-attenuated vaccine within 30 days before the first dose of study intervention. Administration of killed vaccines and mRNA or inactivated COVID-19 vaccine is allowed. * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. * Has a history or current evidence of any condition, therapy, or laboratory abnormality or other circumstance that might confound the results of the study, interfere with the participant's participation for the full duration of the study, such that it is not in the best interest of the participant to participate, in the opinion of the treating investigator.