Cope with basic accommodation, but not at first (4)
I believe the answer is:
hack
'cope with' is the definition.
(if you can hack something, you can cope with it)
'basic accommodation but not at first' is the wordplay.
'basic accommodation' becomes 'shack' (basic type of building).
'but not at first' means to remove the first letter.
'shack' with its first letter taken off is 'HACK'.
(Other definitions for hack that I've seen before include "Journalistic drudge (for the chop?)" , "try to access unauthorised data" , "Journo" , "Cut roughly - routine writer" , "mediocre writer" .)