Final answer:
A drooping arch of the soft palate with deviation of the uvula to the unaffected side is likely caused by a lesion in the glossopharyngeal or vagus nerve, which impairs the function of the soft palate muscles on one side.
Step-by-step explanation:
The condition described where the arch of the soft palate droops and the uvula deviates to the unaffected side is typically caused by a lesion affecting the cranial nerves. Specifically, it could be damage to the tenth cranial nerve, also known as the vagus nerve, or the ninth cranial nerve, referred to as the glossopharyngeal nerve. These nerves are responsible for the motor innervation of the muscles of the soft palate. When one side is damaged, the muscle on that side cannot contract properly. As a result, when the patient attempts actions such as swallowing or saying "ah," the soft palate elevates asymmetrically, and the uvula points away from the side of the lesion because the muscle on the unaffected side is still functional and pulls the uvula towards it.