Exernal branch of the superior laryngeal
The external branch of the superior laryngeal nerve travels with the superior thyroid artery toward the superior pole of the thyroid. So, it's in the right place to be injured by surgery on the upper pole of the gland. The external branch of the superior laryngeal nerve innervates cricothyroid, the muscle responsible for elongating the vocal cords. The patient's symptoms of horseness and a weak voice would fit with an injury to this nerve and the denervation of this muscle.
The inferior laryngeal and recurrent laryngeal nerves are one in the same--the recurrent laryngeal changes its name to the inferior laryngeal at the inferior border of cricopharyngeus. This nerve innervates all of the other muscles of the larynx, besides cricothyroid. It runs with the inferior thyroid artery to the inferior poles of the thyroid--it could be injured in any surgery involving the inferior thyroid artery or inferior aspects of the gland. The internal branch of the superior laryngeal nerve pierces the thyrohyoid membrane along with the superior laryngeal artery--it is a sensory nerve only. It supplies sensory innervation to the mucosa of the larynx, superior to the vocal folds.
Ansa cervicalis is a branch of the cervical plexus which innervates the strap muscles. It does not innervate any muscles in the larynx.