Cessna 172 pilot reported rough engine condition on climb. Pilot returned to departure airport and landed.

Narrative

All preflight inspections were conducted in accordance with the aircraft checklist with no discrepancies noted. Engine oil quantity was observed below the full mark of 7 quarts but above the minimum refill requirement of 5 quarts. During engine run-up; the magnetos were cleaned in accordance with normal procedures. Subsequent magneto checks were within limits; with an approximate 75 RPM drop on each side. All avionics and systems functioned normally during taxi.A soft-field takeoff was initiated from Runway XXL at ZZZ. At approximately 300 feet AGL; a sudden impact-like sensation was felt; consistent with a possible bird strike. Seconds later; the engine began to run extremely rough; accompanied by a noticeable smoke or oil odor in the cockpit. I immediately requested a short approach and was cleared to land on Runway XXR.During the approach; maximum engine RPM was limited to approximately 2050 RPM with full throttle and mixture. The aircraft was unable to climb or accelerate and remained slow at approximately 500 feet AGL. Upon reducing throttle on final approach; the severe engine vibration subsided.A normal landing was completed on Runway XXR; and the aircraft exited at Taxiway 1 and proceeded via Taxiway 2. During taxi; the engine exhibited significant vibration. Taxi was terminated at maintenance shop after oil was observed actively leaking from the aircraft. Engine shutdown was performed normally in accordance with the checklist.Following engine shutdown; a post-flight inspection was conducted. During this inspection; it was noted that an engine cylinder had blown.