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

Inclusion Criteria: * Must have a diagnosis of platinum-sensitive or platinum-resistant recurrent or refractory epithelial ovarian, primary peritoneal or fallopian tube carcinoma and have progressed, relapsed, or recurred through at least one or more prior lines of standard-of-care therapies. For platinum sensitive patients, the standard of care therapies include additional platinum-containing regimens and bevacizumab * Have been informed of other treatment options * Must be HLA- A\*02.1 and HLA-DP\*04 positive. Retesting is not required for patients who have previous documented positivity * Patient's tumor must be positive by histological or molecular assay for NY-ESO-1 * Have an Eastern Cooperative Oncology group (ECOG) performance status of 0 or 1 * Life expectancy of \> 4 months * At least 4 weeks from prior chemotherapy, radiotherapy or immunotherapy, or prior investigational agents * Must have measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 * Must have adequate venous access for apheresis (pheresis catheter placement for cell collection is allowed) * Since the study drug may affect pregnancy since it targets proteins present during development, women of childbearing potential are requested to use acceptable methods of birth control for the duration of the study and until persistence of the study drug is no longer detected in the patient by polymerase chain reaction (PCR). This may be a period of several years. Methods for acce…