Deer took cover around centre of spinney (4)
I believe the answer is:
hind
'deer' is the definition.
(hind is a kind of deer)
'took cover around centre of spinney' is the wordplay.
'took cover' becomes 'hid' (I've seen this before).
'around' is an insertion indicator.
'centre of' says to take the centre.
The middle of 'spinney' is 'n'.
'hid' placed around 'n' is 'HIND'.
(Other definitions for hind that I've seen before include "posterior" , "Female animal; at the back" , "Be much better than (someone)" , "Female red deer" , "Female beast" .)