Some lessening of cover for the crown (7)
I believe the answer is:
haircut
'some lessening of cover' is the definition.
'haircut' can be an answer for 'cover' (haircut is a kind of cover). I am not sure about the 'some lessening of' bit.
I don't understand the rest of the clue.
Can you help me to learn more?
(Other definitions for haircut that I've seen before include "DA, for example" , "style for shock" , "Pageboy is one" , "Less of a shock after this" , "It proved fatal to Samson" .)