When you banish a unit and replay it to that location at a battlefield with a hidden card, does the hidden card stay if not played?

Yes, the hidden card stays. Here's why:

When you banish a unit and replay it to that same battlefield, control of the battlefield is never lost. The spell/ability resolving creates a chain and the replayed unit goes pending on the chain. Per Rule 187.4.c and the official FAQ clarification (2026-04-29), control of a battlefield cannot be lost while there are items on the chain — the pending unit still on the chain prevents the cleanup from removing your control, and then when the unit finalizes it enters that battlefield, giving you a unit there again.

The hidden card is only removed when you lose control of the battlefield during a cleanup (Rule 107.3.d, FAQ #4628). Since you maintain control throughout this entire sequence, the hidden card remains undisturbed and can be played on a later turn as normal.