Reckless forward, not British (4)
I believe the answer is:
rash
'reckless' is the definition.
(synonyms)
'forward not british' is the wordplay.
'forward' becomes 'brash' (similar in meaning).
'not' suggests deleting specific letters.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'brash' with 'b' taken out is 'RASH'.
(Other definitions for rash that I've seen before include "Foolhardy with skin ailment" , "Hotheaded" , "Precipitate" , "Careless" , "Impetuous; skin ailment" .)