(ぬく) Literally, to pluck out. The mouse plucked the thorn out of the elephant’s paw. But can be used in the sense of “Take this data out of a database.”   TTT
(はずす) To remove temporarily, without destroying:Take the player out of the starting lineup. Take the door off of its hinges.  Can also mean to miss or dodge.   $$$
(はいじょ) Remove a blockage – for instance a tree which has fallen across a road, or a troublesome shareholder who is trying to stop a company’s plan to close a factory.
(じょがい) To eliminate a team from a tournament.