Power-ups can be found at the top of the screen at the start of a race. You can choose which power-ups you want to use (King Sling, Kart Repair, or Boost) before pulling the slingshot back to start the race.
The power-ups first become available after racing against Bomb for the first time in the Champion Chase. If you have already completed all of the events in the first track of Seedway, and you have recruited Stella onto your team, tap the white arrow to the right of the event icons to move to the next track where you can race against Bomb and unlock the power-ups. You can then go back and complete the power-up challenges from the first screen.