The Paddle is an unused mission that took place at some point during Chapter 4. Data pertaining to it was found in the game's data files. This mission was going to involve Jimmy stealing Dr. Crabblesnitch's paddle, then framing Damon for the crime. Most cutscenes, including the introductory one, were either dummied out or never finished in the first place.
Jimmy was to sneak into Dr. Crabblesnitch's office, retrieve the paddle, and sneak out again without being caught. He then had to run to the football field, where the Jocks are practicing, and place the paddle in Damon's gym bag without getting caught.
There are scripts in the game for Algie to distract Damon by throwing something at him, and for Ricky to create a distraction by flirting with Mandy and provoking the Jocks into chasing him. Small cutscenes for Burton breaking up the fight between Algie and Damon, and of Mandy calling Ricky a pervert before the Jocks threaten him, are still in the game. There is also something in the script for Jimmy to be attacked by some Jocks, although there's no information about exactly where, when, or why.
The mission is finished by Jimmy taking a picture of Damon holding the paddle, then showing the picture to Dr. Crabblesnitch.
Note that this video is an edited recreation.