I had another big and small idea.
The big one is: Instead of a random card, the HIT ME is a timeable roulette. Sorted by value, so it's all the 2s, the 3s, ect... Spinning fast enough that hitting a value is easy, but hitting a suit is hard. Flushes essentially stay 1/4, but straights are slow and guaranteed. Four of a kinds get harder over time, as they get removed. Royal flushes mean waiting for a full rotation, and taking the 1/4 chance on suit.
The small one is: When you place a card in your hand, the old one goes to your 'inventory'. That way you can rearrange your hand, and you can't accidentally remove a card.