Although I'm fairly clueless when it comes to puffers (but because I'm a retired microbiologist), to me, that white area around the mouth looks like a bacterial infection.
Certain bacterial infections, can after time, so atrophy the tissue around mouth, that the jaw gets locked and isn't useable.
Somewhat like Columnaris (duck lips) in cichlids.
For Columnaris an antibiotic that works on Gram Negative bacteria is appropriate, but if the disease has progressed too far before treatment is applied, as in some the case above photo, where the fish arrived with the disease, it was hopeless, and euthanization was best for it and to try to contain its spread. These bacteria can be spread in nets, a few drops of water, substrate, equipment, filter media, and fish o fish contact