Flight Instructor reported a rough running engine and loss of engine power during initial climb out. Flight returned to the departure airport.

Narrative

Myself [instructor pilot (IP)] and my student [student pilot (SP)] attempted to conduct a 141 PPL training flight. The aircraft in use was a Diamond DA40NG; based at ZZZ. The tail number was Aircraft X. Following normal procedure; a preflight inspection and run up were completed to SOP standards with no abnormalities noted. No open Squawks" were noted in the airplane's maintenance log. Taxi was without incident. During takeoff no abnormalities were noted in engine performance. [IP] was the pilot flying as there were relatively high winds for a PPL student with less than 10 flight hours. Runway XX was used during takeoff. During the initial climb out; power was set at 100% load and the engine was working normally. At the acceleration altitude of 400 ft the flaps were retracted and to allow the airplane to accelerate to cruise climb speed. Once this speed was reached; the power settings was reduced to 92%. Almost immediately after 92% power was reached; and at roughly 700 ft AGL; the engine began running rough; shaking the airframe. This was accompanied with a perceived loss of power; identified by a drop in the nose; a reduction in airspeed; and a reverse in altitude trend from climbing to descending briefly. Aircraft sink sensation was also experienced by both pilots on board. This engine roughness and power loss persisted for only 4-5 seconds. Engine instrumentation did not appear to reflect what was experienced by the pilots; as the load remained at 92% throughout. As a result of the power loss and high airport traffic flow; [ATC was advised] and a request for immediate landing on Runway XY was made in effort to return the airplane to the ground as swiftly and safely as possible. Following the previously discussed engine roughness; the engine began running normally with no further abnormalities. A landing was made on Runway XX and was unremarkable. The airplane was taxied back to the ramp and was shut down normally with no further engine roughness noted. To summarize; engine roughness was experienced after takeoff in Aircraft X; resulting in [a request for priority handling] and immediate landing. There were no injuries; and the aircraft did not sustain any damage. An after landing inspection did not immediately reveal apparent issues; such as leaking fluids. Additionally; the airplane had 26 gallons of fuel on board indicated with no change."