Will the muscles around the corner of the mouth recover from Botox paralysis? (photos)

From your pictures, the injection weakened the muscle that pulls down on the corner of the mouth (depressor anguli oris). Your injector can actually weaken the other side, as well, in order to re-create symmetry, while you wait for the unwanted effects to wear away. I sometimes perform this treatment on patients that have a "downward" pull on each side of their smile. 

Botox is temporary and your results will gradually go away in a few months. I'm sorry to hear that you've had disappointing results, but your mouth will return to its original shape.

Luckily whether injected correctly or incorrectly the effect of Botox will last 3-4 months, and your facial muscles will return to their original function.  It appears your injector may have injected too medially and superficially - unintentionally treating one of your lip depressors (DAO).  The correct depth of injection for the masseter is too deep to affect the DAO.  The other side can also be treated for symmetry, or you can wait for the muscle activity on the treated side to return.  

It may take months or even up to a year to recover from this. The good news is that you will improve over time. It takes patience and you just have to wait. There is nothing you can do to make it disappear faster, unfortunately but know that overtime things should go back to normal.

The good news is that Botox always wears off and muscle function returns to normal if it was a result of the Botox.  Usually the unwanted effects do not last the entire time of the normal treatments.  Looking in the mirror and trying to force the normal movements sometimes help things improve a bit.  What is important is that you and your injector realize how much was used and exactly where it was placed, so you can avoid a repeat problem.