"Mr. Bun" is the pet rabbit Lil' Oly recruits Razputin to help him chase and catch in the Meat Circus. Raz must protect himself and Lil' Oly from monsters and other enemies, as Lil' Oly seems unconcerned about anything except catching the beloved pet rabbit. Mr. Bun is presumably the very same rabbit that Oleander's father disapproved of so strongly during Oleander's youth.
Catching Mr. BunEdit
As you get higher up into the circus tent protecting Lil' Oly and helping catch Mr. Bun gets more and more difficult. Holding the bunny with Telekinesis should do the trick, but since failing to protect Oly only results in resetting to the beginning of the current challenge, the priority must be on maintaining Raz's mental health.