Flight Instructor with a student at the controls of a Skyhawk 172 reported that the engine began to lose power during climb. The instructor then took control of the aircraft and landed.

Narrative

Report of situation with Skyhawk at FBO on Day 1 at XA:21.100hr and oil change inspection completed Day 0.On a lesson with student reinstating his private pilot license. We have had multiple lessons in a process to refresh his private pilots license and complete and BFR. We had briefed take off emergency and in-flight emergency procedures in every lesson. The student has a cumulative 250hrs total flight time.Aircraft X was preflighted with student and close supervision by instructor going over all systems; fuel quality; and quantity. Preflight notations that co-pilot mic jacks were inop on previous flight and instructor PTT and normal functions of copilot jacks were inop. Instructor to use passenger mic intercom jack and headphone jack. Student insisted he was okay to do all coms even though rusty with coms. He wanted the task load and to get better on coms.Turn coordinator plane portion of gyro noted inop and observed by instructor on previous flight. Ball portion of coordinator properly working and tested proper on preflight. Landing light found inop on preflight due to unplugged under engine cowling. Dash trim and door panel found hanging and loose on preflight. Mag drop observed on run up 100rpm each side and no split RPM between right to left. Carb heat actuation test showed 100 RPM drop and then a stabilization. All systems pass run up test minus comes copilot side coms; and plane portion of gyro in turn coordinator inop. Student and instructor evaluated issues and deemed safe for pattern work only flight and to remain close to the airport.Tower cleared Aircraft X for left closed traffic cleared for take off runway XX; XA:19. XA:20-XA:21 Aircraft X entered runway XX; winds 300-15. Full take off power was achieved at 2500 + RPM; normal take off; and engine gauges in the green noted and voiced between student and instructor. Rotation and climb normal until 3500msl and engine surged; sputtered and stumbled; losing RPM to 1600-1700 RPM; with excessive vibration felt throughout aircraft. Gauges all in green except RPM at 1600-1700. Carb heat was immediately applied with a slight smoothing of the vibration; yet then continued to decrease in take off power. Instructor took the controls and ran quick visual checks of primer; mags; tanks R/L; Mix; throttle; and tried adjusting mixture rich; lean then back rich mixture. Slight climb from 3750 msl gained at vx 65 mph to 3850 during testing upwind; and evaluating risks and landing spots; about to turn crosswind. Continued to try vx climb 60-65 mph; power would not increase; and altitude could not be maintained. Instructor told student to request priority handling with tower. Tower cleared Aircraft X land any runway.Instructor maintained 30 degree bank turn with best glide at 80 mph. Could not make XZ/XA runway due to construction cones and barricades. Aimed and landed mid field runway XY. Bled off speed just above runway with 15 kt tailwind; landed without event. Taxied off with excessive vibration throughout plane. Taxi to FBO under own power with excessive vibration. On taxi the engine oil pressure and oil temperature were noted in the green areas of gauges; no excessive electrical draw on systems. No adverse smells noted at any time in the cockpit. Student and instructor debriefed situation and what possible causes could be. Seatbelts were maintained throughout situation at all phases of flight and CRM was always utilized between student and instructor. Secured aircraft and let supervisors and lead instructor know of the issue.