can you play a hidden card at a battlefield in reaction to your unit moving from that battlefield to base?

Yes, you can react to a card or ability that moves your unit from a battlefield to your base, provided that the move is caused by an effect (spell or ability) and not by a "Standard Move."

According to RiftJudge FAQ #3126:

• If an opponent plays a spell or uses an ability that moves your unit, that card or ability goes onto the chain.
• Once it is on the chain, the state becomes CLOSED, and you have an opportunity to play a [Reaction] card.
• You may reveal your hidden card at that battlefield to react to that spell or ability, provided the hidden card was not hidden on the same turn (see FAQ #9904).

Important Requirements & Constraints:

1. Timing: The card must have been hidden on a previous turn. You cannot play a card from "Hidden" on the same turn you hid it (RiftJudge FAQ #9904).
2. Targeting: If your hidden card requires targeting, it must target a unit (or other valid object) at the same battlefield where it is hidden, unless the card explicitly states otherwise (Rule 811.1.d.2 and RiftJudge FAQ #7786).
3. Standard Moves: You cannot react to a "Standard Move" (a manual move of a unit by a player). Standard Moves do not use the chain, so they never create a window for reactions.
4. Control of the Battlefield: If you react to an effect that moves your last unit away, you must ensure you do not lose control of the battlefield before the hidden card resolves. Generally, because the item that triggered the move is on the chain, you retain control of the battlefield until the chain resolves and the cleanup occurs, allowing you to use the hidden card (Rule 187.4.c).

Summary of the sequence:

1. Opponent plays a card or ability that targets and moves your unit: The item goes on the chain.
2. State becomes CLOSED: You are now in the window to play a [Reaction].
3. You reveal your hidden card: You place it on top of the chain (it now resolves before the opponent's move effect).
4. Resolution: The chain resolves in Last-In, First-Out (LIFO) order. Your hidden card resolves first, followed by the opponent's move effect.