Final answer:
CN VII innervates the 2nd pharyngeal arch, which may be related to the patient's symptoms.
Step-by-step explanation:
The nerve that innervates the 2nd pharyngeal arch is CN VII, also known as the facial nerve. The 2nd pharyngeal arch gives rise to several structures in the head and neck, including the muscles of facial expression, the stapedius muscle in the middle ear, and the stylohyoid and posterior belly of the digastric muscles. So, in this case, the patient's recurrent infections and mucus discharge on the surface of the neck may be related to dysfunction of CN VII.