C152 pilot reported an engine malfunction during takeoff climb resulting in a return to the departure airport.

Narrative

On DATE at approximately XA:45 pm local time; Aircraft X; a Cessna 152; experienced rough engine and partial power loss 300 ft. into the initial climb after taking off on Runway XX at ZZZ under VFR conditions under Part 61 flight training. The airplane was successfully landed back on Runway XX with no damages or injuries to the airplane and pilots.The power loss was felt at 300 ft. During takeoff; full throttle was applied with full mixture and no carburetor heat. Everything registered normal until the airplane got airborne. The RPM on the tachometer was reading below normal levels at around 1;900 to 2;000 RPM instead of the usual 2;400 RPM during takeoff. Due to the fix-pitch propeller; a lower angle of attack was implement to shallow out the climb and build airspeed. The airplane was able to achieve a maximum airspeed of 75 kts. in straight and level flight. No further attempts were made to continue the climb once the airplane reached 500 ft. Troubleshoot procedures were conducted in compliance with section 3 of the Cessna 152's Pilot's Operating Handbook for Engine Failure During Flight (Restart Procedures) and Precautionary Landing with Engine Power.After executing the troubleshoot procedures; normal RPM levels did not register on the tachometer. The RPM levels remained low. There was also a strong heavy vibration that came from the engine; especially when the carburetor heat was applied. The vibrations felt like a 'skipping heartbeat'. It was suspected that one of the cylinders did not provide enough power; possibly due to a fuel; valve; or timing issue with the engine. Fortunately; there was no indications of a total engine failure no smoke; no burning smells; and no visual indicators of abnormal exterior irregularities.ZZZ Tower was then notified of Aircraft X's issue on climbout where a priority landing was requested. ZZZ Tower issued several airplanes to give way to Aircraft X as it circled to land back onto the Runway XX. After landing; ZZZ controllers were informed about why Aircraft X needed to have priority back to the Runway. Then Aircraft X was taxied to the Maintenance Hangar at FBO where the A&P mechanics were informed of the incident.