Good on you for entering the competition, and making something unique.
My suggestion for making Phoune Clicker truly one-button accessible, would be to forget about moving the mouse pointer anywhere. Just left-click or SPACE BAR to play.
Then make clicking do more interesting things, with things happening after every 10 presses or so. Make them really clear. That would make a one-switch / one-button accessible activity to fit in with the competition rules. Still worth doing, I'd say.
Best wishes,
Barrie
OneSwitch.org.uk