Final answer:
The fleshy mound at the medial corner of the eye is called the lacrimal caruncle, which is near the tear drainage system but not directly involved in it.
Step-by-step explanation:
The fleshy mound at the medial corner of the eye is called the lacrimal caruncle. It is situated near the lacrimal fluid drainage system, which includes the lacrimal bone forming the anterior, medial wall of the orbit, with a shallow depression called the lacrimal fossa extending into the nasolacrimal canal. This canal allows the excess tears to drain into the nasal cavity. The lacrimal caruncle itself is not involved in tear drainage but is a small, reddish conical tissue at the corner of the eye that contains glands producing oils and other substances contributing to the tear film.