Flight Instructor on training flight with student flying a C-172 aircraft reported engine failure during departure climb. Returned to airport uneventfully.

Narrative

Myself and my student departed Runway XX at ZZZ. After rotation I noticed the fuel mixture was not in the full rich position; and enriched the mixture. We flew an extended upwind following traffic for a northbound departure. As we began our Northbound turn at approximately 2600 feet msl with full power set. Our engine coughed and began to shake as the RPM reduced to 1000 rpm. I took the flight controls from my student and began a right turn towards the approach end of Runway XY as I pitched for best glide. I notified ZZZ tower that we had a rough running engine and were returning to land on Runway XY. My student assisted me and we completed the 'Engine Failure During Flight' checklist. This had no effect. Our engine power remained around 1000 rpm at full throttle and was extremely rough running. I then [requested priory handling] with ZZZ as my student squawked XXXX. I tried multiple power settings and was able to achieve a very shaky 1600 rpm. We lined up to land on the runway and completed the 'precautionary landing with power' checklist. As we touched down on Runway XY our engine shutdown completely. We were able to roll off the runway at Taxiway 1 where we received assistance from ground crew to transport the airplane.