You mentioned Apollo was the God of healing, does that include psychological healing aswell?
Personally, I do believe so but it's largely based on my own interpretations and reasoning.
Apollo is the god of health and health does not end with our physical body. The fact that he's been associated with philosophy, getting to know oneself and bettering oneself (as I understand it, as a human being, as a whole) also prompted me to think that his area of influence over health extends to mental health as well, on the contrary to his son, Asclepius, who was primarily associated with medicine and the more physical aspects of health. We could search for answers in ancient sources, however, the ancients did not have the same terminology for such things as we do nowadays.
"These two, Asklepios and Plato, did Apollo beget;
One, that he might save the soul; the other, the body."
Olympiodorus, Life of Plato, 2.164 - citing the alleged inscription on Plato's tomb
This could technically indicate that by Apollo's will, Asklepios helped with physical wellness, while Plato was supposed to "heal the soul" - perhaps Apollo's contribution to healing the psyche (soul & mind) could be interpreted as him assisting with mental wellness as well.
On top of that, Apollo had functions as the god of religious healing and ritual purification, so maybe this could also be linked to healing the psyche.
But how about we move on to some mythology now?
"As for Aristaeus, after the death of Acteon, we are told, he went to the oracle of his father, Apollo, who prophesied to him that he was to change his home to the island of Ceos and told him likewise of the honours which would be his among the Ceans."
Diodorus Siculus, Bibliotheca historica, 4.82.1
Aristaios was a rustic god whose son Acteon fell to the fatal wrath of Artemis, punished for having seen her bathing. Such an event is obviously a huge tragedy and experiencing it would leave any parent terribly heartbroken.
In this case, Apollo essentially gave advice to his son Aristaios, who'd just suffered the loss of his child, to move to a different place where he might feel better, and perhaps we could even assume, find it easier to recover.
[Although this is just me interpreting this fragment of the text in the context of the possible connection between Apollo and mental health.]
Apollo is also the god of music and art, both of which can be quite therapeutic. The god was believed to play songs for the Moirai to help them relax, for example.
Now let's take a look at another text referencing mythology:
"Neither doth Thetis his mother wail her dirge for Achilles, when she hears Hie Paeeon, Hie Paeeon."
Callimachus, Hymn to Apollo, 17
Here we can see that hearing a pean (a song or lyric poem expressing triumph or thanksgiving sung during some festivals dedicated to Apollo) had the power to soothe the grief of Thetis, the mother of Achilles who was mourning the death of her son.
On a side note, if you're interested in other gods who could be associated with mental health Hygeia, daughter of Asclepius and Epione, goddess of good health was sometimes thought to be the giver or protectress of mental health.
