A Flight Instructor reported engine failure and landing aircraft in a field.

Narrative

Aircraft in training flight experienced an actual engine failure upon go-around procedure during a simulated engine failure and power-off 180 to land at ZZZ. On climbout; Student Pilot in training applied full power; flaps cleaned and pitch was set at Vy around 72kts. At 400 feet AGL; aircraft lost engine performance and immediately Instructor Pilot took controls. Aircraft was too low to pull out QRH and immediate action was decided by both Pilots to commit to an off-field landing. Instructor Pilot turned 90 degrees to the left looking for a best spot to land. Area was surrounded by open terrain; mostly flat plains filled with brushes. Upon touchdown; Pilots called out 'brace for impact' and landed hard on the mains before lowering the nose wheel.Aircraft rolled about 500ft to a complete stop on the field. Both pilots worked together to ensure checklist Engine Shut-Down was secure before exiting the aircraft. Area was very remote from any civilization. Student Pilot was able to use handheld radio to transmit to aircraft in the vicinity of the practice area. Once contact was made; Student Pilot addressed off-field landing and requested to relay information to Departure Airport ZZZ1 and the FBO. Post landing walk-around; airframe; propeller and landing gears had no significant visible damage. Local Sheriff and A&P mechanics arrived on site 3 hours afterwards which conducted further inspection of the engine. It was concluded that engine failure occurred due to push rod on one of the cylinders was completely bent.