SR-20 flight Instructor reported a message that stated a cylinder problem appeared prior to landing but after following the checklist; the message was gone. Upon roll-out; the engine failed and the aircraft stopped partially on the runway. The flight Instructor and Student pilot coordinated pushing the aircraft off the runway; restarted the engine; and taxied to parking.

Narrative

10 minutes prior to landing at ZZZ a CHT CAS message appeared for CHT on Cylinder #1. The checklist called to reduce power; increase airspeed; and increase mixture. The CHT CAS message disappeared. I kept the mixture at the new setting while returning to ZZZ to land. After landing on [Runway] XXL at ZZZ my student rolled down the runway with the power at idle. Approaching Taxiway 1 and turning to exit; my student applied throttle to complete the turn and exit the runway at [Taxiway] 1. The engine started coughing and died and we rolled just past the runway edge marking and came to a stop well before the hold-short lines. I radioed the Tower Controller to let him know that our engine died - he replied that he understood and asked that we get out and push the airplane past the hold-short line to exit the airplane. I also heard the Tower Controller tell the aircraft on final to sidestep to the parallel runway for landing. The controller also told a business jet that their departure on [Runway] XXL would be slightly delayed due to a disabled aircraft still on the runway. After these radio calls I informed the Tower Controller that we would be getting out to push the aircraft past the hold-short line. As the instructor I exited the aircraft and pushed while I instructed my student to steer the aircraft. It took an estimated 3 minutes to push the airplane past the hold-short line. Once past the hold-short line I boarded the aircraft and attempted a hot start procedure - which was successful with a leaner than normal mixture setting and received clearance to taxi to parking without further problems. Our maintenance provider determined during their engine run that the engine will stall when advancing the throttle from idle with a full rich mixture; boost pump on; and air conditioning on. They also advised that with normal ground leaning that the engine appears to be running normally. Cirrus's SR20 POH does not have a high altitude ground leaning procedure. Their start procedure calls for the mixture to be set to full rich which is not ideal for operating at higher altitudes especially in ZZZ1. This incident occurred after landing where the throttle was at idle during the after landing roll-out; and because of the CHT CAS message 10 minutes prior to landing the mixture was more rich than normal - but not full rich. I believe that the student's application of power to exit the runway may have been too abrupt and with the throttle at idle with a high mixture setting; may have caused this engine failure.