Provide small space, initially, to sleep (4)
I believe the answer is:
doss
'sleep' is the definition.
(I've seen this before)
'provide small space initially' is the wordplay.
'provide' becomes 'do' (I've seen this in other clues).
'initially' indicates taking the first letters.
The initial letters of 'small space' is 'ss'.
'do'+'ss'='DOSS'
'to' acts as a link.
(Other definitions for doss that I've seen before include "Sleep (in cheap lodgings?)" , "easily accomplished" , "Sleep like a tramp" , "go to bed" , "Sleep somewhere convenient" .)