The Amarillo Kid is ?.
A long time ago, the Amarillo Kid challenged Private to a game of mini golf, back when Private was one of the best mini golf players and was known as Mr. Tux. Private quit after the game, which ended in a tie, because he had to hit a girl possum's ice cream cone to tie it up.
Several years later, he arrived at the Central Park Zoo (in th episode Mr. Tux) to challenge Private to a game to prove he's the best in Mr. Tux. At first, Private refuses to fight him. Enraged by this, the Kid locked down the entire zoo and threatened to blow it up with a thermonuclear reactor. Private battles him, and though the game is close, Private wins with the help of a cricket he had saved early. Ashamed, the Amarillo Kid left.
He made another appearance in Showdown on Fairway 18. He asks Private for help when a pair of gophers chase after him for his shell (which he lost in a golf match as a wager). He teams up with Private to "borrow" the Westchester Putter which he will use in a rematch against the gophers. However, he reveals later on that he meant to steal the putter to save his shell. With a feeling of guilt, the Amarillo Kid changes his ways and ends up retrieving the Putter.