Tecnam P2012 pilot reported loss of engine power and trim control on initial climb and returned to the airport.

Narrative

Extremely hot day. Last flight of the day. Looking to the storms we would have to navigate around on the way home. We had already briefed the last takeoff time we could accept given that we were both going to exceed 13 hours anyway; but were trying to stay under 14. Previously during the day; we had started with a static takeoff on the 1st flight where power was fine; rolling takeoff where power on the right engine had started more at around 83%; then increased as the takeoff increased; and a 3rd static where the engines achieved power in the 90s on static run up; then normal power as takeoff developed. On all these takeoffs we had variable power readings; and the takeoff performance was as expected. We had only briefly discussed these numbers; but was consistent with previous experience and information sharing among pilots that power numbers varied considerably on takeoff.Earlier in the day both engines had TLO (indication) lights. I was told by Maintenance Control they were being tracked. Those lit intermittently during the day.In this takeoff; we had [Runway] XXL with over 7;000 ft. feet available and a static run up had us at around mid 80s % power on static run up. The FO was Pilot Flying. We took note; and I told him to release the brakes. I noticed a somewhat sluggish takeoff roll; but we were also heavy and it was very hot. We rotated the airplane and I noticed it was not climbing normally. The sluggish climb and low power indications persisted. At this point I knew it wasn't a sensor issue but a real power issue. No CAS (Crew Alert System) or EECS (Electric Engine Control System) lights illuminated. Climb rates varied from about 100 to about 450 feet per minute at full power.Also; during the climb; the stall warning was activating at a high airspeed. When the airspeed reached about Vy; the stall warning would go off. At this point was a choice to ignore the training of reacting to a stall warning and fly to Vx; or to ignore the Air Data Computer information to react to a stall indication. I chose the get as close to the stall warning as I could while setting it off intermittently. While this warning went off several times; our airspeed was never in the stall regime as displayed on the speed tape. We didn't even get to Vx. We were struggling to climb the airplane to the assigned altitude. We advised ATC that we were coming back and they cleared us to 2000 and vectored us back to [Runway] XXL. During the climb we saw the power increased to the mid 90s in %; then shortly before I reduced power; it showed a decrease again. I was keeping the power full forward since the takeoff roll. I eventually reduced power when we were close to short final. During this I also ran out of nose down trim; while descending and not in the yellow arc. I recall the airspeed at the time I ran out of trim at about 163 knots.We landed without further incident and taxied in to the gate. Passengers were deplaned. My FO wrote down some engine indications during the issue. Climbout - 85%/81% @ 400 ft.Cruise - 95%/95% 47'/49' @ 2000 ft.Descent (full power) - 82%/84% 45 inches/48 inches @ 1300 ft. - my FO noted this reading was taken right before I intentionally reduced power for landing.During the flight my FO noticed some small amounts of black smoke from the left engine; but I did not see that when I looked back.