C172 Flight Instructor with student reported while climbing to exit icing conditions near the aircraft service ceiling the aircraft engine failed. The instructor descended and the engine eventually developed power so they returned to departure airport.

Narrative

While flying with an Instrument student on an IFR flight plan we began to pick up unforecast trace icing. We requested a climb to 12;000 feet and were granted a block from 10;500 to 12;000 to try to get out of the icing. When we attempted to climb to 12;000; we suddenly began to lose power and began an uncommanded decent. I took control of the aircraft and [requested priority handling]. I was briefly able to see the ground to my right and chose to descend below the clouds into VMC but below ATC's MIA (Minimum IFR Altitude). I confirmed that the Carburetor Heat was on; that I would be able to maintain VMC and chose a course I could visually fly to ZZZ (our intended destination). We leveled off at about 7;000 feet MSL in rugged terrain. Upon level-off; the engine 'coughed' briefly and then began producing full power. ATC provided us with advisory vectors and we continued the flight to ZZZ. I returned the controls to the student who flew the ILS XX Circle-to-Land.Prior to departing; the student and I had done a thorough preflight briefing. There were no AIRMETs for icing. And the Icing encountered was extremely light. Upon landing; we reviewed both the initial preflight briefing and obtained another preflight briefing with an intended route of ZZZ to ZZZ1 then ZZZ2. Based on this brief and our visual observation we decide to depart ZZZ on an IFR flight plan while working to remain clear of clouds. The student was flying. Shortly after departure we determined that this would not be possible on the route to ZZZ1 so we diverted to direct to ZZZZZ intersection for the approach into ZZZ2. En-route; we requested and received multiple deviations to avoid clouds. Shortly before reaching ZZZZZ; we were forced to transition through a cloud. Prior to entering the cloud; I checked weather at ZZZ3 and ZZZ2 - both were reporting Clear below 12;000 and I felt we would not be in the cloud for long. We again picked up trace icing but were easily able to maintain altitude until we came out the other side of the cloud. We then had a sudden loss of nearly 500 RPM on the engine and again had an uncommanded descent. We were in VMC but headed back into the clouds. ATC issued me an altitude warning. We were unable to comply with their instructions to climb. We were also directly over and in visual contact with ZZZ4 airport. I took control of the airport and was focused on maintaining control and VMC. The engine then 'coughed' again and began developing power. I then chose to cancel IFR and fly VFR up the valley because I could see that we could remain VMC back to our home base at ZZZ2.ATC was fantastically helpful and patient throughout the emergence. True professionals. I was asked to call the ZZZ Center on the phone and had an excellent; professional discussion with the Supervisor there.After discussing this flight with the FBO Owner and a more experienced CFI-I we determined that the likely issues were inadequate leaning with the Carb heat on and operating near the Ceiling for the aircraft.