Started to have a bit of a bet, backing a horse (5)
I believe the answer is:
began
'started' is the definition.
('begin' can be a synonym of 'start')
'a bit of a bet backing a horse' is the wordplay.
'a bit of' says to take the initial letters.
'backing' is a charade indicator (letters next to each other).
'a horse' becomes 'egan' (I can't explain this - if you can you should give a lot more credence to this answer).
The initial letter of 'bet' is 'b'.
'b'+'egan'='BEGAN'
'to have' is the link.
Can you help me to learn more?
(Another definition for began that I've seen is " Commenced, started".)