If you are playing in the Events or Daily Challenges, click on the 'Try again' arrow at the bottom of the screen to restart that particular game.
If you are playing in regular game mode, and are mid-game, use the 'Undo' button to go back to the move you wish to change or go back to the beginning. Using Ctrl+Z on a keyboard does make this go faster. If you click the 'New Game' + sign at the bottom of the screen your current game will end and you will start a new game that is different than the one you are currently playing.
If you are playing regular game mode, and have reached the No more moves! screen. Select 'OK' and then 'Try Again' in the Game Over screen.