Not a branch of cervical part of Facial artery:
The cervical part of the facial artery runs upwards on the superior constrictor of pharynx deep to the posterior belly of the digastric, with the stylohyoid  and to the ramus of the mandible.
The cervical part of the facial artery gives off the ascending  palatine, tonsillar, submental, and glandular branches for the submandibular salivary gland and lymph nodes.

Reference: BD Chaurasia Volume 3, ed 6 page no 102
Ascending palatine artery
Tonsillar artery
Submental artery
Muscular artery