Go to the doctor (ortho) and rule out anything major. Odds are it's not really anything serious but there is a problem and it needs fixed. Step one is to rule out anything medically.
Step two is check out the mechanics. Just because you pay for lessons doesn't mean you are getting quality instruction. I have no clue who is giving the lessons but make sure they are reputable and know what they are doing. To be honest I don't see how he can really learn anything in a half hour session once a week. Is he doing drills, is he throwing and the guy critiques, is he breaking stuff down to understand...... Typically you get lessons for an hour and if the lesson starts at 6:00 they want you loose at that point. They don't want to spend part of the hour waiting on you to get loose unless it's early and they are teaching that aspect to you.
Step three is once he's healthy then let him pitch. If he's taking pitching lessons then he needs to pitch. A half hour lesson and then to pitch here and there is not going to allow the arm to build up the strength and endurance to actually pitch. It may be a case the occasional ramping up without any regularity that's causing discomfort through inflammation. This is why you need to see a doctor.