C172 instructor reported a rough running engine resulted from a stuck valve.

Attributes
ACN	1685883
Time
Date	201908
Local Time Of Day	0001-0600
Place
Locale Reference	ZZZ.Airport
State Reference	US
Environment
Flight Conditions	VMC
Light	Daylight
Aircraft 1
Make Model Name	Skyhawk 172/Cutlass 172
Operating Under FAR Part	Part 91
Flight Plan	None
Component
Aircraft Component	Cylinder
Person 1
Function	Instructor
Qualification	Flight Crew Commercial Flight Crew Flight Instructor Flight Crew Instrument
Experience	Flight Crew Last 90 Days 180 Flight Crew Total 600 Flight Crew Type 580
Events
Anomaly	Aircraft Equipment Problem Critical Deviation - Procedural Clearance

Narrative:

While performing a checkout flight with a low time ppl I had the pilot perform an engine out drill by reducing the power to idle and informing the pilot the engine had failed. The drill was completed satisfactorily and I informed the pilot to recover at 1;000 feet MSL and climb to 2;000 to perform landings at [a nearby airport]. Shortly after the smooth application of full power the engine stumbled and began shaking. I took over controls of the aircraft and attempted to troubleshoot systems. All engine gauges indicated normal. The engine continued to run but with reduced power. I [notified ATC] and determined I was able to climb at approximately 100-200 fpm. I continued climbing at vy until I was within gliding range of ZZZ; proceeded inbound at 3;000 feet MSL; spiraled down once positioned above the runway; and landed without issue. The aircraft was able to taxi under its own power to transient parking. It was determined the aircraft had a stuck valve.

Original NASA ASRS Text

Title: C172 instructor reported a rough running engine resulted from a stuck valve.

Narrative: While performing a checkout flight with a low time PPL I had the pilot perform an engine out drill by reducing the power to idle and informing the pilot the engine had failed. The drill was completed satisfactorily and I informed the pilot to recover at 1;000 feet MSL and climb to 2;000 to perform landings at [a nearby airport]. Shortly after the smooth application of full power the engine stumbled and began shaking. I took over controls of the aircraft and attempted to troubleshoot systems. All engine gauges indicated normal. The engine continued to run but with reduced power. I [notified ATC] and determined I was able to climb at approximately 100-200 fpm. I continued climbing at VY until I was within gliding range of ZZZ; proceeded inbound at 3;000 feet MSL; spiraled down once positioned above the runway; and landed without issue. The aircraft was able to taxi under its own power to transient parking. It was determined the aircraft had a stuck valve.

Data retrieved from NASA's ASRS site and automatically converted to unabbreviated mixed upper/lowercase text. This report is for informational purposes with no guarantee of accuracy. See NASA's ASRS site for official report.