Encourage support for British boom (7)
I believe the answer is:
burgeon
'boom' is the definition.
(thesaurus)
'encourage support for british' is the wordplay.
'encourage' becomes 'urge on' (urging on is a kind of encouraging).
'support for' means one lot of letters go next to another.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'urgeon' after 'b' is 'BURGEON'.
(Other definitions for burgeon that I've seen before include "produce sprouts" , "Produce buds - flourish" , "sprout" , "Shoot" , "bud" .)