Student Pilot reported a loss of engine power and a diversion to another airport.

Narrative

Flight Lesson out of ZZZ [airport] to ZZZ1 [airport] for landings; then continued on for maneuver work in the airspace. Departed ZZZ at approximately XA:45. Refresher flight for me with my instructor; (IP) on board. After concluding the lesson; set up for the ZZZZZ arrival at 3;500 ft. We completed the climb to 3;500 ft; contacted approach and finished setting up for our arrival. Soon after; we noticed oscillating RPM's and Load. We also noticed a faint smell of smoke. We received no annunciations on our flight display during the entirety of this event. Noticing this; we had a positive exchange of flight controls. The instructor was in charge of flight controls and communications while I ran the appropriate QRH/QRC Checklists. Instructing pilot [requested priority handling] with ZZZ Approach and told them we would be conducting a landing into ZZZ1 [airport]. At this time; we were approximately 5 NM north of the airport. ZZZ Approach advised us to Squawk VFR; Frequency changed approved to ZZZ1; 2 in the pattern". We initiated the Engine Troubleshooting checklist from our QRH. An element of this checklist instructed us to give the aircraft max load. Upon doing this; we lost about 50% engine power. The Instructing pilot noticed this and realized we were unable to reach certain load performance. During this; we began our descent into the ZZZ1 traffic pattern entering on the left downwind for Runway XX. With 2 planes in the pattern; the instructing pilot [advised] on the ZZZ1 Unicom and advised of a priority landing. There was traffic on upwind and downwind; both planes extended these legs to allow us to safely execute our landing. While in the Traffic pattern; we noticed we lost positive engine control and the instructing pilot performed a power off landing. This was conducted on base to final at approximately 1;300 ft MSL. Landing was made safely. Once on the ramp; the propeller continued to oscillate from 1-10% load. RPM's fluctuated from 500-900 and Oil pressure was reading 1.4 and unstable. Once landing was made; we advised clear of the runway on Unicom and attempted to contact ZZZ ATC. Upon parking on the ramp; we began to see the smoke and safely powered down the plane and secured it. We noticed the entire underside of the plane was covered in oil from the engine all the way to the tail. We contacted our Chief Instructors along with the maintenance team."