SR20 instructor pilot reported an engine control malfunction which resulted in engine power being unresponsive and aircraft controllability issues during a training flight. The flight landed safely as a precaution and taxied to parking.

Narrative

Instructor performed a power-off stall to demonstrate to the student pilot (SP). Upon stall recovery; when full power was applied from idle power; the RPM/Engine did not initially respond after the power-lever was positioned full. After 8-10 seconds; power shot back up to 75% from 0% and held 2500 RPM. Power then remained at 2500 RPM despite any change made back to reduced or idle power - a very abnormal situation. Instructor made the decision to return to ZZZ and keep power at mid-ranges; switched fuel tank and began troubleshooting. Instructor took controls and focused on flying the plane (which was difficult to control altitude and unpredictable; if not unreliable and uncertain as to how the engine/propeller would behave). Instructor and SP ran RPM Warning/Propeller Governor Failure in QRH between Location A and ZZZ1. SP briefed the arrival and ran checklists while Instructor flew the plane and took communications. Ultimately; the decision to declare urgency or an emergency to ZZZ Approach or ZZZ Tower was not made and routine calls were made. Inbound ZZZ1; Instructor pulled power to descend the plane; but the plane was descending very minimally. While aware to both Instructor and SP; Aircraft X accidentally penetrated a ZZZ2 shelf that borders the ZZZ Delta Airspace while flying the aircraft to the best of Instructor ability. The incursion was caused by the inability to maintain a set altitude. Set up on short-final Runway XX at ZZZ; Instructor put flaps in and reduced power to idle; however; power remained abnormally at 1900RPM and 40% for 20 seconds roughly 300 ft AGL. The crew tried to set the plane down but the plane resisted losing power and descending. Mixture was set lower; and airspeed was bled to 75kts with full flaps. The plane was landed safely; power finally dropped below 1900RPM after roughly 30 seconds from when the power lever was initially put to idle; and Aircraft X was taxied back to parking spot XX at ZZZ parking.