Hey John,
I'm not sure if managers are allowed to ask specifically why you need a day off. I'd say you could mention it after you've gotten the job, or bring it up during the interview? Or is it possible to see the psychiatrist during hours you're not actually working?
If they ask, you could say you're in treatment for something, but not bring it up. Or say "depression" or something like that, which is fairly common and more socially acceptable and easier for people to understand?
Wish you much luck!
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