Speculation I rejected with fury as anathema (4,5)
I believe the answer is:
bete noire
'anathema' is the definition.
('bete noire' can be a synonym of 'anathema')
'speculation i rejected with fury' is the wordplay.
'speculation' becomes 'bet' (bet is a kind of speculation).
'i' becomes 'one' ('one' can be a synonym of 'i'**).
'rejected' is a reversal indicator.
'with' means one lot of letters go next to another.
'fury' becomes 'ire' (I've seen this before).
'one' backwards is 'eno'.
'bet'+'eno'+'ire'='BETE NOIRE'
'as' is the link.
(Other definitions for bete noire that I've seen before include "Object of aversion" , "Pet hate" , "undesirable" , "Person or thing one particularly dislikes" , "One particularly disliked" .)