Final answer:
The external branch of the superior laryngeal nerve is at risk of injury during thyroidectomy due to its proximity to the superior thyroid artery and vein. It innervates the cricothyroid muscle, affecting vocal cord tension and speech.
Step-by-step explanation:
The branch of nerve at risk of injury during a thyroidectomy due to its proximity to the superior thyroid artery and vein is the external branch of the superior laryngeal nerve, which is a branch of the vagus nerve. The nerve is at risk because it often lies close to the superior thyroid artery and vein, which are associated with the thyroid gland. The external branch of the superior laryngeal nerve innervates the cricothyroid muscle, which helps in modulating the tension of the vocal cords during speech.