My half of child’s horse (3)
I believe the answer is:
gee
'my' is the definition.
(I've seen this before)
'half of child's horse' is the wordplay.
'half of' means to take half the letters.
'child's horse' becomes 'geegee' (I've seen this before).
'geegee' cut in half is 'GEE'.
(Other definitions for gee that I've seen before include "American exclamation" , "- - whiz! (US)" , "Well I never" , "Golly" , "Gosh" .)