Psychiatrically means in a way that has to do with the study and treatment of mental illness. If a report says someone is psychiatrically unwell, it means a psychiatrist would see a problem of the mind, not a broken arm. “Psychiatrically speaking” means “if we look at this the way psychiatry does.” It is a long, formal adverb. You will almost never hear it in a shop or a café; it belongs in medical notes, court papers, and textbooks. Students sometimes reach for it as a fancy way to say “crazy.” That is not what it does, and it is unkind besides.