Give up, almost entirely (4)
I believe the answer is:
quit
'give up' is the definition.
(both can mean to resign)
'almost entirely' is the wordplay.
'almost' means to remove the last letter (most of the word but not all of it).
'entirely' becomes 'quite' (I've seen this in another clue).
'quite' with its final letter taken off is 'QUIT'.
(Other definitions for quit that I've seen before include "Go away from or abandon task" , "Leave - resign" , "Decamp" , "Leave one's post" , "Leave or abandon" .)