Cover overdue taking a short time (8)
I believe the answer is:
laminate
'cover' is the definition.
(I know that laminate is a more specific form of the action cover)
'overdue taking a short time' is the wordplay.
'overdue' becomes 'late' (similar in meaning).
'taking' means one lot of letters goes inside another.
'short time' becomes 'min' (short for minute; 'short' indicates an abbreviation).
'a'+'min'='amin'
'late' enclosing 'amin' is 'LAMINATE'.
(Other definitions for laminate that I've seen before include "Anti-male (anag.)" , "Material constructed in layers" , "Material made from layers" , "Material made up of bonded sheets" , "Thinly cover" .)