Step 1: Recall the nerve supply of the ear canal.
The skin of the external ear canal gets sensory supply from several nerves: the auriculotemporal branch of the trigeminal nerve (V) in the front and upper wall, the auricular branch of the vagus nerve (X), also called Arnold's nerve, mainly in the floor and posterior wall, and small contributions from the facial (VII) and glossopharyngeal (IX) nerves.
Step 2: Connect the vagus nerve to the cough reflex.
The vagus nerve also carries sensation from the larynx, trachea, and bronchi, the same pathway that triggers the cough reflex when these areas are irritated. Because Arnold's nerve is a sensory branch of the vagus, touching or cleaning the ear canal skin it supplies can send signals along the vagus that trigger the same cough reflex, a phenomenon called Arnold's reflex or the ear-cough reflex.
Step 3: Rule out the other options.
(A) Trigeminal nerve (V): Supplies sensation to part of the ear canal but is not linked to the cough reflex pathway.
(B) C1, C2 innervation: The great auricular nerve, from C2 and C3, supplies the outer ear (pinna) skin, not the classic cough-triggering pathway of the canal.
(D) Facial nerve (VII) branches: Give a small sensory contribution to the ear canal but are not the nerve responsible for the cough reflex.
Step 4: Final Answer.
The cough response on cleaning the ear canal is mediated by the vagus nerve, cranial nerve X, through its auricular branch (Arnold's nerve).
\[ \boxed{\text{The X cranial nerve}} \]