Start of affair with next American goddess (6)
I believe the answer is:
athena
'goddess' is the definition.
(I've seen this before)
'start of affair with next american' is the wordplay.
'start of' says to take the initial letters.
'with' is a charade indicator (letters next to each other).
'next' becomes 'then' (I've seen this before**).
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
The initial letter of 'affair' is 'a'.
'a'+'then'+'a'='ATHENA'
(Other definitions for athena that I've seen before include "One-time goddess" , "Greek goddess of a capital city" , "wise but fierce goddess" , "Greek goddess of wisdom etc." .)