SR20 flight instructor reported engine malfunction inflight. Returned to departure airport.

Narrative

Departing Date at near XA00 Local time; instructor and student proceeded to the local practice area operating as normal with no apparent problems. After about 30 minutes of maneuvering in the practice area; when configuring the aircraft for slow flight by reducing power; apparent engine roughness; changing rpm; and lack of power control occurred at lower power settings (approximately starting at 35% power and below.) (Prior to this maneuver the student remarked about the power being unusually difficult to adjust) The Pilot in Command took flight controls upon recognizing the issue with the student; and began troubleshooting solely off of systems knowledge by increasing the mixture to full and trying various other mixture settings; which all yielded the same roughness. The Pilot in Command turned the aircraft into the direction of ZZZ while continuing to troubleshoot the issue with no improvement; which lead to an immediate return to ZZZ. The pilot did not [request landing priority]; but advised ZZZ Tower of the issue expecting traffic sequencing priority. Tower gave permission and winds allowed for landing the inactive Runway to make a straight in approach. The Pilot in Command maintained a higher approaching altitude and airspeed towards the Runway as a precaution for the possibility of a total power loss. The pilot established the aircraft on a high altitude final approach in order to avoid making a normal approach at low power where engine roughness occurred. Once landing was assured the power was reduced for landing and flaps added resulting in a safe landing. Once the power was reduced to idle over the runway and the aircraft slowed while pulling off of the runway the engine quit running and prop stopped similar to a feathering prop. Once off of the runway the engine was restarted for taxiing successfully; but still ran rough at lower power. Aircraft X was then properly squawked and sent to maintenance. Recommend more thorough maintenance flight procedures are established to test aircraft such as Aircraft X; which have been in long term storage or maintenance prior to be released to students. Also recommend; following long term storage or maintenance a set amount of dual or maintenance flight hours are should be put on an aircraft until it is released for student solos.