A Tracon Controller reported a military aircraft deviated from their assigned route and flew below the Minimum Vectoring Altitude.

Narrative

Aircraft X was conducting a special procedure. I instructed Aircraft X to depart ZZZZZ heading 030 [at] 6;000 [ft]. Aircraft X read the instruction back to me. However after the procedure; the aircraft did not go to point ZZZZZ as published; they flew a 340 heading. When they called on departure; they informed me they were heading 340 for an own navigation visual approach. This is not what they were instructed to do. This heading takes them directly towards the 5;600 MVA. I informed them to remain on the heading due to the slow climb of the aircraft. However they proceeded to reach the 5;100 MVA sooner than expected and I needed to turn them to avoid the 5;600 MVA. They were instructed that they were entering a 5;600 MVA and no delay to 6;000; turn left heading 270. They were at 5;300 in the 5;600 MVA and turning to a 270 heading. I would recommend not leaving them on the heading they took upon themselves to fly on departure and give them a low altitude alert for the MVA they were about to enter.