Flight Instructor reported engine failure shortly after takeoff. Aircraft returned to field and landed without incident.

Narrative

Engine startup was completely fine; nothing out of the area during taxi either. In the runup area I noted a faint popping that was rhythmic while engine power was at 900 RPM. Aware that this might be an issue; I was focused on the runup to ensure that nothing else seemed out of the ordinary. The runup check was fine; confirmed by myself and my student; well within 150 RPMs of the expected drop. Both magnetos were also within 50 RPM of one another. After our runup check; my student talked through our standard takeoff briefing. The plan was: Cut power; brake; and taxi off if power was lost during the ground roll. Cut power; land; brake; and taxi off if power was lost after liftoff; but with enough runway remaining to land and stop (negligible time at ZZZ). Pitch for best glide; use any remaining power to fly to a landing area straight ahead if power is lost below 500 ft. AGL Pitch for best glide; bank 45 degrees to the right; use any remaining power available; and attempt to land on Runway XX if power was lost above 500 ft. AGLOn the initial takeoff roll the power rose to an appropriate level for takeoff; more than 2300 RPM. There was no indication during the ground roll that an aborted takeoff was needed. The early portions of the initial climb seemed normal; then the power suddenly dropped to something below 2000 RPMs; per our memories. At this point; I took over the flight controls from my student and started to assess the situation. Mixture was full rich; throttle was full; and I made the decision based on the remaining power to return to land on Runway XX and [advised ATC]. During the turn back to Runway XX I had a minimal instrument scan so I don't recall a lot of airspeed; altitude; or power indications during that portion of flight. At one point; the stall warning horn indicator came on and I lowered the nose to maintain airspeed; but did not note what speed I was holding after the stall warning horn stopped. Throughout the turn it appeared that we didn't have sufficient power to maintain a climb. I was almost exclusively flying the plane visually and scanning for potential places to land if we did not have enough energy to get back to the runway due to the low altitude at which we experienced the drop. I did not get a reply from Tower after the call; and they had cleared a departing 172 for takeoff. As I was lining up for Runway XX Tower cleared us to land and advised the traffic to abort the takeoff and exit the runway immediately; which thankfully they did. At this point I threw in whatever flaps we had; cut the power; and came in for the landing. We had excess energy that I think was the result from nosing down after the stall horn; so the touchdown was fairly fast and flat near Taxiway XX. As soon as we touched down I applied maximum braking and was able to stop by Taxiway XY. We still had some power so we were able to taxi back to the ramp and shut down.After these issues; Maintenance discovered that the cause was a failed magneto; and some potential issues with the spark plugs in cylinder three. The magneto was replaced as well as the spark plugs.My thoughts throughout the whole process were mainly on deciding whether or not to plan for a forced landing straight ahead; or trying to turn back. I decided to turn back based on the power that we did have; and the lack of landing options ahead of us. In the past; I had talked with another Instructor about his experience where he also lost power at ZZZ1 and he was able to return to land under partial power and below the predetermined minimum safe altitude from our typical pre-takeoff briefings. I have also previously practiced engine failures from a takeoff attitude at altitude and knew that the altitude loss we had experienced then was about 500 ft. with zero power. Based on all of this I decided the best choice would be to try to attempt to turn back to Runway XX. From that point on; after I committed to the turn I was scanning for landing areas and obstructions while attempting to maintain a pitch that would minimize our descent rate without inducing the stall warning horn. Once I was in a position where I had the airport made; my attention turned to planning on landing safely with the traffic that was on the runway. I lined up between Runway XX and Taxiway XZ; concerned that I may have to land on the taxiway to avoid the aircraft that was in the ground roll if they could not abort the takeoff. Once they cleared the runway; I did whatever I could to get the plane down with flaps and power.I do not believe there was anything that could have been done to prevent this issue from occurring beforehand; based on the normal magneto check and performance in the takeoff roll and early stages of climb. Our Maintenance staff had been following the recommended service times on the magnetos as well as spark plugs; and did a great job analyzing and correcting the issue afterwards. ZZZ Tower did a phenomenal job clearing the runway and preparing for our landing. The same is true for the pilot who made the exit from the runway shortly into his roll; and I believe he was a student pilot solo. At the end of the day; the reason this was a favorable outcome came down to proper training for all involved.