Keen on horse that's not first (4)
I believe the answer is:
into
'keen on' is the definition.
(if you're 'into' something you're keen on it)
'horse that's not first' is the wordplay.
I cannot really see how this works, but
an anagram of 'not' is 'nto' which is within the answer.
'first' could be 'i' (I in Roman numerals eg Elizabeth I) and 'i' is found in the answer.
This accounts for all the letters.
This may be the basis of the clue (or it may be nonsense).
Can you help me to learn more?
(Other definitions for into that I've seen before include "Curious about" , "Four this twelve is three" , "A fan of" , "digging" , "Taking an active and lively interest (in something)" .)