Sturgeon's party covers one cut (4)
I believe the answer is:
snip
'cut' is the definition.
(I know that snip is a type of cut)
'sturgeon's party covers one' is the wordplay.
'sturgeon's party' becomes 'snp' (I can't explain this - if you can you should give a lot more credence to this answer).
'covers' is an insertion indicator.
'one' becomes 'I' (Roman numeral).
'snp' enclosing 'i' is 'SNIP'.
Can you help me to learn more?
(Other definitions for snip that I've seen before include "Cut with scissors in small strokes" , "Steal" , "Cut - price bargain" , "Cut with shears" , "Bargain - crop" .)