A TRACON Controller conducting OJT reported their trainee vectored a climbing aircraft for traffic below the MVA.

Narrative

Training in progress. Trainee released 2 C208 aircraft off of ZZZ1 airport. These aircraft are assigned 255 heading and 4000 ft. altitude. Trainee then released a C510 off of ZZZ2 airport on 260 heading to 4000 ft. I thought the trainee was going to climb the C510 above the 2 departures then turn north. He ended up changing his plan at the last second and climbing the first C208 to 6000 ft. and stopping the second at 3000 ft. Once the C510 got to 4000 ft. he turned it northbound. This had him heading for a 6000 ft. MVA. The trainee ended up getting distracted by other aircraft and wasn't climbing the C510. I instructed the trainee to climb the C510. He still wasn't climbing fast enough so I told him to turn it west away from the MVA. The trainee gave him a heading westbound but it still did not end up being enough and the aircraft entered the 6000 ft. MVA below 6000 ft. I should have stepped in earlier and over keyed the trainee. This particular trainee has a lot of experience so I chose to tell them what to do instead of keying the frequency myself but that ended up being the wrong decision. I also found out later that the trainee was having severe back pain and ended up going home sick.