If opponent uses Unchecked Power, can I let damage resolve and then use Hidden Card?

No, you cannot. You must play your hidden card before Unchecked Power resolves.

According to RiftJudge FAQ #7958, the window to play a hidden card as a reaction exists while Unchecked Power is on the chain, before it resolves. Once Unchecked Power resolves and deals its damage, there is no further opportunity to react until a new chain begins. If your units die from that damage, you have missed your window to act.

Here is the sequence of the game state:

1. Chain Starts: Your opponent plays Unchecked Power. It goes on the chain.
2. Reaction Window: You have priority to play a reaction. This is your opportunity to flip your hidden card. If you do, it will be on the board when Unchecked Power resolves and will be subject to its damage (12 damage).
3. Resolution: If no further reactions are added, the chain resolves (LIFO). Unchecked Power deals 12 damage to all units.
4. Combat Cleanup: Any units that took lethal damage are sent to the trash.
5. No Post-Damage Window: There is no window to play a hidden card after the damage has already resolved and potentially cleared the battlefield. If you wait until after the damage resolves, your window has closed.

This is consistent with Rule 358.3, which governs the legality of playing cards based on timing and state. You cannot use a hidden card to "dodge" the damage of a spell that is already resolving.