Cessna 172 Flight Instructor with student reported engine roughness while maneuvering. Flight crew diverted to a safe landing. Maintenance discovered a stuck exhaust valve.

Narrative

I departed ZZZ1 airport to the northwest to conduct flight training with a low hour student pilot. I was well rested that day and in good spirits. We started by heading near the Harbor off the coast at 4500 ft. MSL varying altitude working on slow flight; power off and on stalls; and fundamentals of flight. Weather was VFR with few and scattered clouds reported earlier in the day ranging in bases from 2500 - 4500 ft. with a broken and/or overcast layer visible farther north toward ZZZ. After completing the upper air maneuvers we practiced a descent down to a lake and ran the descent checklist. I had my student begin to prepare for a ground reference maneuver over an area of the lake and instructed them to level off at 2500 ft. MSL while positioning to setup the maneuver correctly. Once facing south my student added power and we experienced immediate engine roughness and noted a loss of RPM. I took control and started trouble shooting by establishing best glide; flying toward an emergency landing spot; and running checklists. The engine at this point was still operating and generating no greater than 2000 RPM allowing me to hold my altitude of 2500 ft. MSL and 80 KIAS. Realizing the engine was still working in some capacity; I contacted ZZZ Approach who I had already obtained local flight following from and let them know I was experiencing engine roughness and would attempt to return to the closest airport; that being ZZZ. I chose my route while holding altitude to stay over low terrain and areas that would provide for an acceptable off-field landing while going direct as possible to the field. The route took me over the highway from the lake and then over the Harbor and farm land before reaching ZZZ. I maintained altitude in case we had a full engine failure to allow altitude to make a field or ZZZ once we got closer. I was immediately cleared to land upon entry to ZZZ's airspace and landed with no other issues. After landing; I taxied to transient parking to shut down and secure. Upon inspection the next day by Maintenance; it was discovered that a stuck exhaust valve was the reason we had experienced low compression. The whole cylinder head was replaced and the rest of the engine of was examined after the event. This was my first experience with an engine issue in flight and was an eye opening experience. I do not feel that my reaction time to the event was diminished but there was an air of uncertainty to whether this would become a complete engine failure. In the moment I did not feel panic and addressed the situation as best I could; calling upon training experiences and knowledge I had obtained prior in my flight training. The one thing I would go back and do differently would be to have complete clarity in my language; I announced engine roughness and my intentions to return to ZZZ; I will remember in any future event to announce clearly [requesting priority handling]. The next day I was off duty and the days up to writing this report I have been replaying the events and wondering what I could have done better. This event though startling has not deterred me from continuing to fly as it's something I have a great passion for and have wanted to do since I could first remember. I feel immense gratitude for the support from my employer and coworkers at the flight school as well as my flight instructors who trained me to this point. This has affirmed in me my need to continue to keep my skills fresh and practice good ADM; thinking about how to achieve the best outcomes possible in future situations. I believe the maintenance team at our school does a great job maintaining and inspecting aircraft to keep them in a safe and airworthy condition. I think the failure of the engine component is nothing that could have been prevented. I will continue to instruct and run thorough preflight and run ups before takeoff to ensure we avoid future scenarios like this as best as possible.