Paychecks were rewards of 
20 given every twenty minutes for playing Adopt Me!, as the check reads. It appears to be from the "Bank of Adopt Me", and is signed by NewFissy & Bethink.
Once a player clicked "Cash Out", 20 were added to the player's Bucks count. The paycheck popup would then disappear, to return in the next twenty minutes.
On September 22, 2022, during the jobs update, this feature was removed from the game.
Trivia
- Paychecks used to be able to be stored in inventories, but they now no longer can be.