C172 flight instructor with student reported experiencing engine issues with the RPMs dropping as low as 1000 and cycling back to 2000 several times. The instructor took over the controls and decided to return to the departure airport. Reportedly; the cause of the engine issue was a bent push rod.

Narrative

This event occurred in Aircraft X during a lesson with a student. After rolling wings level from a clearing turn engine RPM dropped to 1000. After five seconds the RPM increased to 2000. The RPM kept cycling in this manner with no change from throttle; mixture or pitch. I believed a complete engine failure was possible; so I took command of the aircraft from the student; and started flying back to ZZZ since it was the closest airfield. I pulled the carb heat on; verified the RPM was full and mixture rich; primer locked; master on; fuel on both. I contacted ZZZ Approach on XXX.X and [requested priority handling] to ensure I would receive priority handling and services if an off field landing would be required. I was able to make it back to the field and performed a straight in landing for Runway XX. I contacted ZZZ after safely landing and ensured I no longer required assistance. The cause of the partial engine failure was a bent push rod. No pre-flight action nor run up would have caught this condition. It was a good reaction to make an immediate return to the field.