Crooked mouth. What caused this, and what can be done? (Photo)

It is difficult to tell from this photo. Consider though, that we all have assymetries, both static and dynamic. That is, some people have assymetries at rest and some have them when they are moving. We all have assymetries in muscular tone which can contribute to the condition you highlight here. It is analogous to having one arm stronger than the other. Causes can include congenital imbalance, and as you have already mentioned, a Bell's Palsy type situation. Skeletal imbalances can also contribute to a crooked mouth. The best way to get to the bottom of this is to consult with a local plastic surgeon

Based on your photo it appears you have some facial bony asymmetry. The easiest correction would be some filler and if there is any nerve weakness some Botox to weaken the normal side to correct the asymmetry. An in person consultation is best

There is another cause which is a possible mild Bell's Palsy but that is not the question Danimal.  The question is how to fix it.  The photo you show is not really adequate to determine what the problem is as far as the asymmetry goes.  I would suggest a consultation with a board certified plastic surgeon to discuss your options both temporary and longer lasting before doing anything.  Good luck, Dr. Schuster in Boca Raton.