Half-puss and half-Manx cat (4)
I believe the answer is:
puma
'cat' is the definition.
(puma is a kind of cat)
'half-puss and half-manx' is the wordplay.
'half' suggests halving the letters.
'and' means one lot of letters go next to another.
'half' indicates taking half.
'puss' with half the letters taken is 'pu'.
'manx' cut in half is 'ma'.
'pu'+'ma'='PUMA'
(Other definitions for puma that I've seen before include "International rugby player" , "Large American feline resembling a lion" , "Large wild American feline" , "Large American cat" , "American wild cat" .)