Almost broke cover (4)
I believe the answer is:
skin
'cover' is the definition.
(skin is a kind of cover)
'almost broke' is the wordplay.
'almost' means to remove the last letter (most of the word but not all of it).
'broke' becomes 'skint' (I've seen this before).
'skint' with its final letter removed is 'SKIN'.
(Other definitions for skin that I've seen before include "Hide - flay" , "Flense" , "Personal cover?" , "Outer layer of a fruit or vegetable" , "Outer layer of tissue" .)