Bet impression includes church forerunner (10)
I believe the answer is:
antecedent
'forerunner' is the definition.
('antecedent' can be a synonym of 'forerunner')
'bet impression includes church' is the wordplay.
'bet' becomes 'ante' (ante is a kind of bet).
'impression' becomes 'dent' (dent is a kind of impression**).
'includes' indicates putting letters inside.
'church' becomes 'ce' (abbreviation for Church of England).
'ante'+'dent'='antedent'
'antedent' enclosing 'ce' is 'ANTECEDENT'.
(Other definitions for antecedent that I've seen before include "Preceding in time or order" , "prior" , "Something logically preceding another" , "Going before (in time/order)" , "Something existing before another" .)