Atkinson but not with horse (4)
I believe the answer is:
john
'atkinson' is the definition.
(I know that atkinson given name John)
'not with horse' is the wordplay.
'not' becomes 'jon' (I can't explain this - if you can you should believe this answer much more).
'with' is an insertion indicator.
'horse' becomes 'H' (abbreviation as in horse power).
'jon' going around 'h' is 'JOHN'.
'but' acts as a link.
(Other definitions for john that I've seen before include "Magna Carta king" , "where Jack came from" , "Barleycorn, Bull or Doe" , "Fourth gospel" , "American novelist, d. 1968" .)