Answer

I didn’t understand why any special discussion is needed. Suffering, by definition, is a mental state, and depression is a clear case of suffering.
Why would a person who is suffering be allowed to shower with hot water? Is it really permitted to violate a prohibition just because someone is suffering a little? The halakhic decisors wrote that in a case of suffering one may be lenient when the water was heated before the Sabbath or by a solar water heater. See Shulchan Arukh, Orach Chayim 326:1, and Rabbi Akiva Eger there.