Fleet Street admitting endless trouble and strife (5)
I believe the answer is:
swift
'fleet' is the definition.
('swift' can be a synonym of 'fleet')
'street admitting endless trouble and strife' is the wordplay.
'street' becomes 'st'.
'admitting' is an insertion indicator.
'endless' means to remove the last letter.
'trouble and strife' becomes 'wife' (Cockney rhyming slang).
'wife' with its last letter taken off is 'wif'.
'st' placed around 'wif' is 'SWIFT'.
(Other definitions for swift that I've seen before include "Fleet" , "Jonathan ..., clergyman and satirist" , "flying" , "Fast - novelist" , "Fast; bird; writer" .)