Right in attempt to obtain private capital (10)
I believe the answer is:
bridgetown
'capital' is the definition.
(I've seen this before)
'right in attempt to obtain private' is the wordplay.
'right' becomes 'r' (common abbreviation).
'in' indicates putting letters inside.
'attempt to' becomes 'bid'.
'obtain' becomes 'get' (synonyms).
'private' becomes 'own' (I've seen this before).
'r' placed into 'bid' is 'brid'.
'brid'+'get'+'own'='BRIDGETOWN'
(Other definitions for bridgetown that I've seen before include "capital" , "in Caribbean" .)