PA-28 Flight Instructor on training flight reported loss of engine power after takeoff and returned to departure airport.

Narrative

At XA:16 am local time my student and I experienced partial power loss. The lesson for the day was TOL's and we were planning on staying at ZZZ. We both performed our preflight inspection and the only issue that we found was that the aircraft was not topped off. I called for fuel and the fuel truck came and topped off the aircraft. We both checked the fuel and found no issues. The aircraft also had 7 quarts of oil. Winds when we received the ATIS were out of 110 at 11 knots and the runway in use was Runway X. My student and I started up the plane and went through our after start checklist and taxi checklist and taxied over to B1. My student called ground and received our taxi instructions. Ground had us taxi to Runway X via B;E. We performed our runup in the runup area to the right of Runway X. Everything in the runup was normal no issues. Both mags showed a drop of 100-125 rpm. The student then performed our pre-takeoff brief; and we said that if we would have an engine failure with no runway remaining that we would shoot for the highway or Runway XX if altitude permitted. We performed our before takeoff checklist in the runup area but we did not go mixture full forward. I have my student wait until we are number one; and I have them announce lights; camera; action before we roll out on the runway so that we are not waiting for a prolonged time with mixture forward. (Lights are landing light and strobe; camera is our mixture and action is to look for traffic one last time). We then pulled up to Runway X at A1; and we were number 1. Tower cleared us for takeoff on Runway X and told us to make right traffic; and gave us a wind update. Winds were out of 100 or 110 at 10 knots gusting 16 knots. My student went through lights; camera; action and we pulled out on the runway stopped and increased power to 2000. All of our engine instruments were in the green. My student then released his brakes and I had him perform his call outs (airspeed alive) and check his rpm to confirm that it was between 2300-2400 rpm. There were no issues and we rotated at 60 knots and climbed out at 76 knots. We turned right crosswind and as we were turning right down wind I heard our rpm decrease as we were reaching 1000 ft. I thought the student pulled back power; but he advised me that he had not. I then said 'my controls;' and took control of the aircraft. I checked our rpm and we were still getting 2450-2500 so I thought that maybe I had a fouled spark plug or failed magneto. Tower cleared us for a touch and go; but I advised Tower that I wanted a full stop. Tower then cleared us to land Runway X. I then had my student read the before landing checklist as I performed it. I kept the approach close to the airport; and essentially performed a power off 180 in case we would loose power if it turned out to be something more serious. I landed the aircraft with no issues and I had my student perform his after landing checklist. Tower directed us to taxi to Runway X via A; and I advised tower that we wanted to taxi to the runup area. Tower then said to continue to Runway X via A and taxi to the runup area at pilots discretion. In the runup area my student and I ran the Cleaning Fouled Spark Plug checklist. We then checked the magneto's and had 125 rpm drop on each magneto. I then ran the whole runup checklist again and found no issues. I then had my student perform the before takeoff checklist and after he was complete I advised him that we were going to pay close attention to our rpm on takeoff roll. I then told the student that if we would have abnormal rpm on takeoff roll that we would abort takeoff and that if we had an engine failure with not enough runway left we would shoot straight ahead for the highway or Runway XX; if altitude permitted. We then pulled up to Runway X at A1 and again we were number one. Tower cleared us for takeoff on Runway X and to make right traffic. I then had my student perform lights; camera action again and we lined up on centerline; increased power to 2000; and all of the engine instruments were in the green. We released the brakes and added full power. We checked our rpm on the roll and we were seeing 2350 rpm. I then told the student that everything looked normal and to continue. On the climb out when we reached 300ft AGL we had the power cut out so we only had partial power but this time it was more abrupt. I took controls immediately. The plane was still climbing but very slowly. I continued to climb in case we would have a engine failure to give us extra gliding range. I was unsure if it would make it back to Runway X so I asked tower if we could land on XX even though I knew we would have a tail wind. Tower cleared us to land on XX and I started a close right base for Runway XX. At this time we were around 500 ft. so I added full flaps and tried to slip but our airspeed increased and we floated down the runway I knew that if we were going to land we had to do it soon because we were using up a large amount of runway; and if we waited too much longer we would not make the runway. I tried to force the plane down but we were going too fast. We bounced and I decided to go around. I added full power but the plane hesitated for a couple seconds (longer than normal). The power then came in and I retracted flaps to 25 because I knew flaps full would make it more difficult to climb and we already had a partial power with a tail wind. As soon as we went around I advised Tower that we were going around and that we were requesting priority and were going to land on Runway X. We might have had slightly more power but it was hard to tell it may have just been from having flaps 25 in. I was watching our airspeed very closely because I did not want to stall. Once clear of our obstacles I retracted our flaps one notch at a time. We got to right around 300ft. and entered a close right downwind/base for Runway X and landed. We were able to taxi off the runway and back to the ramp with no assistance.