C172 Flight Instructor reported as power was increased aircraft shook and failed to get full power resulted in a priority return to departure airport.

Narrative

I am a flight instructor with 100 hours of dual given. The flight took off at ZZZ using runway XX. We turned southeast at 700 feet. We stopped climb at 1;000 feet for traffic. After spotting the traffic; we maintained visual separation from the aircraft and then continued to climb to 2;000 feet. Once at 2;000 feet; I instructed the student to introduce full power and climb to with a pitch up to 20 degrees while use rudder to mantain coordinated flight. Then I instructed my student to pull the power out to idle and lower the nose while use rudders to maintain coordinated flight. The student quickly adjusted the throttle in a swift motion. The student was aggressive when he retard the throttle to idle. I mentioned to not be so aggressive. After stabilized in the decent I instructed the student to repeat the maneuver. Once again the student aggressively pulled back the throttle to idle. Again I instructed the student not to quickly pull the throttles back. Finally; we repeated the maneuver for the last time. After having the throttle back at idle; I instructed the student to add normal cruising power and pull up to straight and level. Once he tried to add throttle; the aircraft began to shake abnormally and violently. I took controls for the student and turned the aircraft back to the airport (ZZZ). The aircraft experienced a partial lost of power. I told ATC that we were coming back to the airport. They told us to make left traffic for runway XX and I told them I was going to land runway XY. That approved it and I landed straight in runway XX. I maintain altitude until I could ensure a power off landing was possible. After landing I told ATC that no further assistance was needed and I taxied back into the airport under the aircraft's own power. Afterward the maintenance team took a thorough look at the aircraft.