C172 Flight Instructor reported a rough running engine and loss of power while returning to the departure airport.

Narrative

I was on a training flight with my CFI applicant when we experienced engine roughness enroute back to ZZZ for closed traffic training. After both concluded that the engine sounded 'off' I took controls. Almost immediately after doing so; the aircraft began to show signs off an issue. After a moment; I [requested assistance] that we were experiencing engine roughness with ZZZ Tower. We were told to proceed directly to the numbers(RWYXX) and cleared to land. I as PIC attempted to climb. However; when adding power; the aircraft responded with more roughness. I then did my absolute best to find a power setting that ran the 'smoothest' while attempting to climb/maintain altitude; adjust the power; locate the airport; communicate with ATC; assess engine out scenarios; check instruments. During this critical event; [student] assisted with navigation; checklist usage; transponder coding; and best places to land. Thankfully; due to CRM and training; we made it safely to ZZZ and landed on RWYXX with about 400 ft to spare.