Enterprising type coming right across river (10)
I believe the answer is:
adventurer
'enterprising type' is the definition.
(a brave or enterprising person)
'coming right across river' is the wordplay.
'coming' becomes 'advent' (advent is the arrival or coming of something).
'right' becomes 'r' (common abbreviation).
'across' indicates putting letters inside.
'river' becomes 'ure' (river in Yorkshire).
'advent'+'r'='adventr'
'adventr' placed around 'ure' is 'ADVENTURER'.
(Other definitions for adventurer that I've seen before include "Drake perhaps" , "One loving dangerous exploits" , "An opportunist" , "one to take risks" , "One off in search of exciting experiences" .)