A Flight Instructor reported loss of engine power while descending to their destination.

Narrative

At the beginning of our descent into ZZZ airport; leaving 11500 for 9500; I was marking up my airport map for all the closed taxiways and runways to ensure we didn't turn down one of them when I felt the nose rising. I thought we were leveling off; but it was too high and then I heard the autopilot give an airspeed warning. The student pilot (SP) was by then going full power; then full mixture. I took the controls; disconnected the autopilot; and ran through the engine troubleshoot flow. Still losing altitude I [advised ATC] when asked by ATC while leaning the engine a bit to see if I could get more power as it was around 1500rpm then back to full rich when there was no increase. A few seconds after the troubleshoot flow we gained some power back to around 2100rpm and ATC noticed we leveled and even had a slight climb at 74kts; but it still did not feel like I had full power.At this time SP did a great job of then running through the checklists himself and helping squawk and input the radio frequencies so I could focus on the flying with no input from me to do so. I had a small discussion with the SP to confirm that to confirm that the runway I was looking at was the longer runway at ZZZ and not a nearby airport to help find the smaller runway we were expecting to land on. Around this time ATC was asking what runway we wanted to land on to which I said any; and for radar vectors to again ensure we were heading toward the correct runway. I believe they also asked if I could land on runway XXL but by then I knew I was lined up for XYR and knew I could glide in if we lost power decided I'd rather treat it as a power-off landing than risk not being able to make it by flying the pattern. At this point; I employed a forward slip and a steep spiral 360 to lose altitude and landed on XYR exiting at taxiway 1. We were able to taxi on our own power and noticed they had rolled the trucks.Once we arrived at the FBO airport operations met with us. I gave him a quick rundown of what happened. I then called Dispatch after the aircraft was chocked and reported to them what happened. They arranged a hotel while I talked to the FBO about tying down the plane. They informed us they could not as they had nothing to tie it to but that they would triple chock it and monitor it and that if the winds picked up would move it to a safe area; or that they could see if they had room in the hanger for an extra charge but that they did this all the time and it would be fine. I held off on moving as they said I could call if I wanted to and after later checking that we were on the tail end of the wind and it would be calm all night decided to not move the aircraft as it would ideally be moved to a maintenance hanger in short time. I believe that it most likely was the engine-driven fuel pump due to us gaining power after the engine troubleshoot flow but I can't say for sure as it all happened in a short time and with a usable runway in gliding range and having already [advised ATC]; I decided to prioritize taking advantage of that for sure landing rather than focus on troubleshooting something that may or may not fix the issue.