Recurrent Adult T-cell Leukemia/Lymphoma

This phase I trial is to find out the best dose, possible benefits and/or side effects of third-party natural killer cells in combination with mogamulizumab in treating patients with cutaneous T-cell lymphoma or adult T-cell leukemia/lymphoma that has come back (relapsed) or does not respond to treatment (refractory). Immunotherapy with third-party natural killer cells, may induce changes in body's immune system and may interfere with the ability of tumor cells to grow and spread. Mogamulizumab is a monoclonal antibody that may interfere with the ability of cancer cells to grow and spread. Giving third-party natural killer cells in combination with mogamulizumab may kill more cancer cells.