C172 Flight Instructor reported engine power loss and returned to the departure airport and made a successful landing.

Narrative

I was conducting an introductory flight in a Cessna 172 in day VFR conditions. I felt like the sound of the power was fluctuating on the climb-out but not enough to decide if it was real or in my head. The RPM gauge had a typical gentle wiggle from turbulence. It was my first time flying this particular 172. I had flown all the others in the flight school by this point but this one. I brushed the sound off as a quirk of the particular airplane. During one phase of the flight I demonstrated to my adept intro student a stable approach descent from 7;500 ft. MSL (2;000 ft. AGL). Carburetor heat on; power reduced smoothly to 1500 RPM; and I established the stable descent at 65 kts. with 30 degrees flaps. I then recovered by about 7;300 ft. MSL by going full power and turning carburetor heat off; and changing flaps to 20 degrees. A few seconds later the engine power started waning; which I was already on alert for. I immediately turned back toward the airport. By the time I completed the westbound turn the engine started shaking and losing power. 400 RPM drops with noticeable shaking every 10 to 20 seconds. I pulled carburetor heat for 30 seconds with no appreciable increase or decrease in power. I called 3 times over Tower frequency and told them I was returning to the airport. I went mixture rich; verified primer locked and fuel selector on both. Mixture rich seemed to help. I adjusted the carburetor heat and mixture several times on the flight back and neither seemed to make any difference. The engine would run smoothly for about 20 seconds and then shake and lose power for 5-10 seconds before coming back to life. I used the available power to climb up to 7;800 ft. until I was within gliding distance of the Runway and glided the airplane back. The airplane ran smoothly after I reduced the power and it kept running all the way in to Parking. I checked the weather conditions after landing and the relative humidity was 32% and the outside air temperature was approximately 65-70 degrees F.