Carboplatin, Gemcitabine Hydrochloride, and Berzosertib in Treating Patients With Recurrent and Metastatic Ovarian, Primary Peritoneal, or Fallopian Tube Cancer

Inclusion Criteria: * Histologically confirmed high grade serous or endometrioid ovarian, peritoneal or fallopian tube malignancy that is metastatic and for which curative measures do not exist. The histology can be confirmed from tissue that was taken at the time of diagnosis. A biopsy at the time of recurrence prior to enrollment on study is not required * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm (\>= 2 cm) with conventional techniques or as \>= 10 mm (\>= 1 cm) with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Patients enrolled in the expansion cohort will be required to have archival tumor tissue available for analysis and be willing to have a tumor biopsy at baseline (after registration and prior to starting study treatment), at cycle 1 day 2 and at cycle 2 day 2. Patients must have platinum sensitive disease and be in their first or second platinum sensitive recurrence. Platinum sensitive disease is defined as recurrence that occurred greater than six months after completion of their last line of platinum based therapy. No non-platinum regimens allowed; prior therapy with PARP inhibitors as well as bevacizumab is allowed * No more than two prior platinum based regimens. One regimen is defined as the interval of treatment from …