C172 Flight Instructor reported engine lost power after initial climb and return to departure airport.

Narrative

Student controls; took off of runway XX with on-course left turn to north approved. During climb circling 270 degrees to the north. We arrived at 3;000 ft.; student followed pitch; power; trim; and starting doing the cruise checklist. As he was starting to lean the mixture. Instructor verified correct. The nose started to pitch up abruptly. I looked down and saw the RPMs around 2200 and what sounded like full throttle. Throttle was not in full position. I took the controls and lowered the AOA while simultaneously running through the checklist. Before I could run the checklist the power dropped RPMs rapidly. I had fuel on both; put mixture full rich; pulled carb heat out; and verified mags on both. After putting the throttle to full and only getting 1500 RPMs and unable to climb; I circled back to the airport telling approach I was inbound for landing. I told them I still had some engine power and I was at 2;700 ft.; so I pulled the power to idle and did a Power off 180 approach. During this I switched to tower and got cleared to land for runway XX.