Better to cover doctrine briefly and wing it (9)
I believe the answer is:
improvise
'wing it' is the definition.
(I've seen this before)
'better to cover doctrine briefly' is the wordplay.
'better' becomes 'improve' ('improve' can be a synonym of 'better'**).
'to cover' is an insertion indicator.
'doctrine' becomes 'ism' (suffix for disciplines or schools of thought).
'briefly' means to remove the last letter.
'ism' with its final letter taken away is 'is'.
'improve' placed around 'is' is 'IMPROVISE'.
'and' acts as a link.
(Other definitions for improvise that I've seen before include "Talk without notes" , "Vamp" , "Quickly make changes" , "Perform without preparation" , "jam?" .)