Rather formal chief fails to finish (4)
I believe the answer is:
prim
'rather formal' is the definition.
'prim' can be an answer for 'formal' (I've seen this in another clue). I'm unsure of the 'rather' bit.
'chief fails to finish' is the wordplay.
'chief' becomes 'prime' (I've seen this before).
'fails to finish' means to remove the last letter.
'prime' with its last letter removed is 'PRIM'.
Can you help me to learn more?
(Other definitions for prim that I've seen before include "Straitlaced and proper" , "Strait-laced - prudish" , "Formally prudish" , "Proper (type of rose?)" , "Schoolmarmish" .)