A Flight Instructor reported the engine cowling came off the aircraft obstructing their vision resulting in a runway excursion after returning to the airport to land.

Narrative

I arrived at the hangar and saw the plane Cessna 172P. I conducted a pre-flight inspection and pre-flight walk-around. The cowling screws were in place; and the aircraft seemed airworthy and intact. The plane was not grounded in the scheduling system we use at the school to track aircraft status. Had it been grounded; I wouldn't have been able to check it out. Since there were no indications of any issues; we proceeded with the flight. The takeoff was normal; and we climbed to approximately 3;000 feet msl when the cowling suddenly came off. My visibility was severely obstructed by the cowling and we began losing altitude. The student; being on his second flight; was mostly a passenger and unable to help guide me to the runway. Given the situation; I [requested priority handling] with ZZZ Approach and returned to the airport. Despite not being able to see the runway; I managed to perform a emergency landing with no injuries; however; due to the obstructed forward visibility; the aircraft stopped in the grass in the runway safety area After landing; I called the Chief Instructor/flight school owner and reported the incident. He informed me that the plane had been undergoing maintenance and should not have been flown. However; there were no signs or indications of this before the flight. According to standard flight school procedure; aircraft under maintenance should be grounded in the scheduling system; preventing instructors from checking them out. This step was not followed; leaving me unaware that work had been done on the plane. I had also flown it the day before without any issues. Flight-ready planes have been kept in the hangar when there is no space on the ramp or to keep them out of severe weather. Previously; I have flown in aircraft that had been parked in the hangar with the proper airworthy status in the scheduling system. In this case; I was able to book and fly the plane solely because it was not grounded in the scheduling system.