Not much is known about the relationship between Stumpy and Eugly, as they barely interact on the show.
Stumpy doesn't talk much with Eugly, but he occasionally does talk about her. Usually, this involves insulting her in some way, for example in "Let's Play Tutus Galore" when he, along with everyone else, laughs at the prospect of Eugly being a ballerina. In "Let's Play With Eugly", Stumpy and Mr. Cat believe that Eugly is causing damage to their friendship with Quack Quack, so they try to break the couple up. When this fails, they both angrily insult her. Eugly manages to remedy the situation by befriending them at the end of the episode.
Eugly sometimes beats Stumpy up on Pretty's orders. However, she rarely does this of her own accord. In "Let's Play Castaways", she decides to rescue him and the rest of the main four from an island. She manages to find him and Kaeloo on one side of the island and carries them to safety, ignoring the fact that Stumpy is chewing on her since he is hungry. In the episode "Let's Play Sheep Vote", Eugly can be seen carrying Stumpy on her shoulders during Mr. Cat's presidential campaign.