Genetically Engineered Cells (NY-ESO-1 TCR Engineered T Cells and HSCs) After Melphalan Conditioning Regimen in Treating Patients With Recurrent or Refractory Ovarian, Fallopian Tube, or Primary Peritoneal Cancer

This phase I trial studies the best dose and side effects of NY-ESO-1 T cell receptor (TCR) engineered T cells and how well they work with NY-ESO-1 TCR engineered hematopoietic stem cells (HSCs) after melphalan conditioning regimen in treating patients with ovarian, fallopian tube, or primary peritoneal cancer that has come back (recurrent) or does not respond to treatment (refractory). The melphalan conditioning chemotherapy makes room in the patient's bone marrow for new blood cells and blood-forming cells (stem cells) to grow. Giving NY-ESO-1 TCR T cells and stem cells after the conditioning chemotherapy is intended to replace the immune system with new immune cells that have been redirected to attack and kill the cancer cells and thereby improve immune system function against cancer. Giving NY-ESO-1 TCR engineered T cells and HSCs after melphalan may work better in treating patients with ovarian, fallopian tube, or primary peritoneal cancer.