I've seen both "study" and "study for" used and I'm not sure which one is more natural. For example:
- https://www.theguardian.com/education/2016/aug/09/whats-it-really-like-to-study-for-a-postgrad-degree-in-europe writes "study for a Master's degree".
- https://www.prospects.ac.uk/postgraduate-study/masters-degrees writes "study a Master's degree"
Also, does the sentence "he's currently studying for a M.A. Media Studies degree at University X" make sense? Or is "he's currently studying for a degree in M.A. Media Studies at University X" better?