During a thyroid operation, a nerve coursing along with the superior thyroid artery is injured. What can be the possible consequence(s)?

The nerve coursing along with the superior thyroid artery is external laryngeal nerve (branch of superior laryngeal nerve, vagus) which supplies cricothyroid muscle.  All the muscles of larynx are supplied by recurrent laryngeal nerve, except the cricothyroid muscle. Sensory supply to laryngeal mucosa above vocal cords is by internal laryngeal nerve and below the vocal cords by recurrent laryngeal nerve.