PA-28 pilot reported drifting off the runway during the landing roll; and incurring minor damage after striking a runway edge light when directing the aircraft back onto the runway.

Narrative

I was conducting flight training at ZZZ airport with my instructor pilot. We focused on practicing takeoffs and landings in the pattern; completing approximately 10 cycles. Afterward; my instructor exited the aircraft and instructed me to perform a solo landing.During this solo landing; I drifted slightly off the runway but corrected the aircraft's path back onto it. At the time; the winds were variable at approximately 3 knots; so it is unlikely they were a contributing factor. However; I cannot confirm whether a sudden gust of wind or another factor caused the drift. I also do not recall whether I made a radio call to the Tower to confirm my reentry onto the runway or switched to Ground frequency.As I corrected back onto the runway; I inadvertently struck a runway light. Upon reflection; I now realize that the safer course of action would have been to shut down the aircraft rather than attempt to re-enter the runway. This would have minimized the risk of a potential propeller strike with another runway or taxiway light.After taxiing back to the hangar area with my instructor; the aircraft's mechanic inspected the plane and identified only minor paint scratches on the wheel fairing. The mechanic removed both rear wheel fairings for further inspection; but noted they had no impact on the plane's ability to fly again safely. My instructor then suggested we return to the air together to practice landings again.