Keypad overlays background apps during search

A couple of quick questions so people can narrow this down: does it happen with the on-screen keyboard docked at the bottom or when it's floating (a small keyboard that can be dragged around)? And is it the Spotlight search, or a search field inside a particular app?

If the keyboard is floating, pinch outward on it (or drag it to the bottom of the screen) to dock it again. Also check Settings > Multitasking & Gestures to see whether Stage Manager is on, since turning it off sometimes fixes overlapping windows. A restart and updating to the latest iPadOS are worth trying too.