Mr. Tench serves as a sort of chorus in the novel, helping to initially develop themes and providing a context for the priest character.
Tench is in a situation that resembles the priest's situation. He feels forlorn and, to some extent, stranded in the southern state of Mexico. He is cut off from his past both physically and temporally. He has known better days and has known some measure of success. Now he is a faded and failed version of himself. All these things are true of the priest as well.
Tench also introduces the priest to the story and helps to provide exposition for the protagonist as a drinker, a man on the run and as a frightened man. The priest's inability to save or deliver himself to safety and his religious impotence are reflected in Tench's professional compromises. He uses sand to form the molds he uses for false teeth:
"I cast in sand," Mr. Tench said. "What else can I do in this place?"