Flight Instructor with student reported the engine sputtered and RPM decreased during climb. The flight crew returned to departure airport and landed safely.

Narrative

At approximately XA20; student and flight instructor departed from Runway XX at ZZZ with intent to operate VFR and conduct practice approaches to Runway YY. Initial climb-out contained no loss of performance; and the ILS YY practice approach was conducted with no incident occurring. ZZZ Approach assigned student and flight instructor an amended climbout instruction of heading 040 and 1;700 [ft.] for missed procedure. Student conducted missed and turned to heading 040 and began climb. At approximately 980 to 1;000 [ft.] indicated altitude; student and flight instructor experienced sudden loss of performance over water and populated area. Engine momentarily lost 100 to 200 RPM and sputtered on climb. Flight instructor immediately took control of aircraft and pitched nose down to regain and maintain safe airspeed; entered a turn toward ZZZ; and immediately [requested priority handling] both verbally and through Transponder Code XXXX. Engine could not maintain normal climb RPM (2200 RPM); so flight instructor had to maintain a relatively lower-than-normal RPM setting for safe return. Flight instructor and student entered right base for Runway XX. Landing was assured at approximately XA43 and was conducted. Flight instructor and student were able to taxi off of Runway XX under own power with no further incidents occurring. No injuries present from either flight instructor or student.