Thinks the first lady's supporting British priest (8)
I believe the answer is:
believes
'thinks' is the definition.
('believe' can be a synonym of 'think')
'first lady's supporting british priest' is the wordplay.
'first lady's' becomes 'eve's' (first woman in the Bible).
'supporting' is a charade indicator (letters next to each other) (in a down clue, some letters go below others).
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'priest' becomes 'eli' (biblical figure).
'b'+'eli'='beli'
'eves' after 'beli' is 'BELIEVES'.
'the' acts as a link.
(Other definitions for believes that I've seen before include "Swallows" , "Credits" , "Accepts as true" .)