My half of horse (3)
I believe the answer is:
gee
'my' is the definition.
(I know that gee can be written as my)
'half of horse' is the wordplay.
'half of' indicates taking half.
'horse' becomes 'gee-gee' (gee-gee is a kind of horse).
'geegee' cut in half is 'GEE'.
(Other definitions for gee that I've seen before include "7th letter" , "Well I never" , "G" , "Gracious" , "Exclamation of surprise" .)