Phase I Study of Recombinant Human IL-15 (rhIL-15) and Mogamulizumab for People With Refractory or Relapsed Adult T-Cell Leukemia and Mycosis Fungoides/Sezary Syndrome

-INCLUSION CRITERIA: 1. Patients must have one of the following histologically or cytologically proven relapsed and/or refractory to at least one line of systemic treatment, T-cell malignancies confirmed by the Laboratory of Pathology, National Cancer Institute (NCI): mycosis fungoides/Sezary syndrome, or adult T-cell leukemia (chronic, acute, or lymphoma subtype by Shimoyama criteria) 2. Patients with luster of differentiation 30 (CD30)+ Mycosis Fungoides/Sezary Syndrome (MF/SS) must have relapsed after or become intolerant to treatment with brentuximab vedotin 3. A formalin fixed tissue block or 15 slides of tumor sample (archival or fresh) must be available for performance of correlative studies. NOTE: Patients must be willing to have a tumor biopsy if prior tissue or adequate archival tissue is not available (i.e., post- enrollment and prior to treatment). 4. Disease must be measurable with at least one measurable lesion by Response Evaluation Criteria in Lymphoma (RECIL 2017) or modified severity-weighted assessment tool (mSWAT) criteria, or have an abnormal clonal T-cell population detectable by peripheral blood flow cytometry 5. Age \>18 years NOTE: Because no dosing or adverse event data are currently available on the use of rhIL-15 in combination with mogamulizumab in patients \<18 years of age, children are excluded from this study, but will be eligible for future pediatric trials 6. Eastern Cooperative Oncology Group (ECOG) performance status less than or equal to 1 (Karnofsky greater than or equal to 80% 7. Patients must have normal organ and marrow function as defined below: * Absolute neutrophil count: greater than or equal to 1,000/mcL * Platelets: \> 100,000/mcL * Total bilirubin: less than or equal to 1.5 X institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST)Serum glutamic oxaloacetic transaminase (SGOT)/Alanine aminotransferase (ALT)Serum glutamic pyruvic transaminase (SGPT): less than or equal to 2.5 X institutional ULN * Serum creatinine: less than or equal to 1.5 X institutional ULN, OR Creatinine clearance: greater than or equal to 50 mL/min/1.73 m2 for patients with creatinine levels \>1.5 institutional ULN 8. Negative serum or urine pregnancy test at screening for women of childbearing potential (WOCBP) NOTE: WOCBP is defined as any female who has experienced menarche and who has not undergone successful surgical sterilization or who is not postmenopausal. 9. Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and 6 months after completion of rhIL-15 and mogamulizumab administration. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. 10. Ability of subject to understand and the willingness to sign a written informed consent document EXCLUSION CRITERIA: 1. Patients with other T-cell leukemias/lymphomas not specified in the inclusion criteria 2. Anti-cancer treatment within 2 weeks of the first dose of rhIL-15 and mogamulizumab (4 weeks for anti-cancer monoclonal antibody or investigational agents, 6 weeks for donor lymphocyte infusion,100 days for allogeneic stem cell transplant) 3. Systemic treatment for acute or chronic graft versus host disease (GVHD) within 12 weeks of the first dose of rhIL-15 and mogamulizumab 4. Cohort 1 (Dose Escalation) only: history of grade 3/4 GVHD, or active grade 1/2 GVHD regardless of treatment 5. Persisting toxicity related to prior therapy of grade \> 1, with the exception of the following: alopecia, sensory neuropathy grade less than or equal to 2, or other grade less than or equal to 2 not constituting a safety risk based on investigator's judgment 6. Patients who are receiving any other investigational agents 7. Current use of immunosuppressive medication, EXCEPT for the following: * Intranasal, inhaled, topical steroids, or local steroid injection (e.g., intra-articular injection) * Systemic corticosteroids at physiologic doses less than or equal to 10 mg/day of prednisone or equivalent; or, * Steroids as premedication for hypersensitivity reactions (e.g., computed tomography (CT) scan premedication) 8. Patients with previous malignant disease other than the target malignancy within the last 3 years with the exception of basal or squamous cell carcinoma of the skin or cervical carcinoma in situ 9. Cohort 1 (Dose Escalation) only: Active or history of any autoimmune disease thought to be unrelated to their malignancy; for Cohort 2 (Dose Expansion), patients with history of autoimmune disease who are not on active immunosuppressive therapy 10. Patients with asthma requiring chronic inhaled or oral corticosteroids, or history of asthma requiring mechanical ventilation. Patients with a history of mild asthma that are on or can be switched to non-corticosteroid bronchodilator regimens are eligible 11. Patients with active bacterial infections, documented human immunodeficiency virus (HIV) infection or positive HIV 1/2 antibodies at screening, polymerase chain reaction (PCR) evidence for active or chronic hepatitis B or hepatitis C, or positive screening hepatitis B virus (HBV)/hepatitis C virus (HCV) serology without documentation of successful curative treatment 12. Presence of uncontrolled intercurrent illnesses including but not limited to ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, cognitive impairment, active substance abuse, or psychiatric illness/social situations that in the view of the Investigator would preclude safe treatment and limit compliance with study requirements 13. Inability or refusal to practice effective contraception during therapy or the presence of pregnancy or active breastfeeding. Because there is no significant preclinical information regarding the risks to a fetus or a newborn infant, all pregnant or breastfeeding woman will be excluded from participation in this trial 14. History of allergic reactions attributed to compounds of similar chemical or biologic composition to rhIL-15 or mogamulizumab, unless felt to be in the best interests of the patient in the opinion of the investigator 15. Patients who received a live vaccine within 30 days of planned start of study therapy. Vaccination with a live vaccine while on trial is prohibited. NOTE: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist) are live attenuated vaccines, and are not allowed