B757 Captain reported APU failure during initial climb and return to departure airport.

Narrative

Aircraft was dispatched with the left generator deferred. FO was PF. On climb-out from ZZZ passing approximately 15;000' the APU faulted with associated load shedding and a small thump that was felt. FO continued to fly. As we ran the checklist; we requested to level at FL200 and to start a turn 'back toward ZZZ;' as I wasn't completely confident that the APU would restart and it would be beneficial to stay in the local area. ATC responded to our request for a turn toward ZZZ by clearing us after a short delay 'to ZZZ;' which we hadn't exactly asked for; but kept us closer to ZZZ. We reset and attempted to restart the APU per QRH procedure. The fault light did not re-appear; yet the APU did not start as expected; observed by continued loss of the utility busses (tried to reset) and APU EGT continually declining on the status page.We contacted Dispatch by VHF; and told our dispatcher that we would be [requesting priority handling] and returning. Workload was high but manageable. Options at this point were ZZZ off our nose at 40 miles; ZZZ1 was 40 miles to our side; and ZZZ2 behind us with a turn required. We chose ZZZ as it was most familiar and also the nearest airport in point of time.We [requested priority handling] and requested to return to ZZZ. Conducted a visual approach to runway XXR uneventfully. Had ARFF inspect the tail of the aircraft upon exiting the runway in case of un-annunciated fire or damage; then returned to the ramp.Physiology for myself and my first officer was good; with me coming off days off and a good rest prior to duty. As a crew; we've flown together several times over the past XX years to include the past X weeks of the bid period; and we work well together. The only thing I would change would be to request 'vectors to stay in the area a minute' vs. 'a turn back toward ZZZ;' which was misinterpreted; but of minor consequence in this case.