From the Super Mario Wiki
A Mini Battle is a battle with a Bandit in Super Mario World 2: Yoshi's Island and its remake Yoshi's Island: Super Mario Advance 3. A Yoshi will fight a Bandit in one of four battles, either Throwing Balloons, Popping Balloons, Watermelon Seed Spitting Contest, and Gather Coins. They are fought in certain levels after a Yoshi gets a key to the Bonus Game shack. Later, they can be played on the level selection screen.
This is Top Secret so LISTEN UP! On the Level Selection screen, hold Select and press X, X, Y, B and A!! (This code is changed to L, L, B, A and R in Super Mario Advance 3)
If the player does so, a screen will pop up listing all the Mini Battles, along with a 2-player version (removed in Super Mario Advance 3) of Watermelon Seed Spitting Contest and a 4-button Throwing Balloons mode.