Gemcitabine Hydrochloride Alone or With M6620 in Treating Patients With Recurrent Ovarian, Primary Peritoneal, or Fallopian Tube Cancer

Inclusion Criteria: * Patients must have histologically confirmed high grade serous ovarian or primary peritoneal or fallopian tube cancer; platinum resistant disease is defined as progression within 6 months after last platinum regimen * 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; measurable disease by RECIST version (v)1.1 with at least one measurable target lesion * Prior therapy: No line limit but no more than 1 prior regimens in the platinum resistant setting; no prior treatment targeting the ATR/checkpoint kinase 1 (CHK1) pathway and no prior gemcitabine as single agent; hormonal therapies immunotherapy, and antiangiogenic therapies (as single agents) do not count as lines; poly (adenosine diphosphate \[ADP\]-ribose) polymerases (PARP)-inhibitors count as a line of therapy unless given in the maintenance setting; PARP-inhibitors given as maintenance after platinum therapy do not count as a line of therapy; prior carboplatin/gemcitabine is allowed provided that there was no disease progression within 12 months after completion of the carboplatin/gemcitabine regimen; subjects may begin protocol treatment at least 4 we…