You get 1 hint per puzzle attempt. As its name implies, it doesn’t show you the exact move, but should nudge you in the right direction. Using it costs you the three-star rating and reduces XP earned.
In Journey, wrong moves just don’t count as a solve – retry as many times as you like, you only lose the “first-try” bonus.
In Rush/Survival, a wrong move is an immediate mistake with no retry, and there’s no skip option.