Well, in the terminal running the `mv` you can do:
1. Press Ctrl+Z.
2. `bg`
And, that will continue the execution of the move in the background.
As a side note; If you suspend/background a lot, then you may want to run `jobs` after the Ctrl+Z and get the number of the `mv` job so you background the right job (e.g. `bg 1`.)
Hope that helps.