Game? Put off, losing heart (4)
I believe the answer is:
deer
'game?' is the definition.
(type of game animal)
'put off losing heart' is the wordplay.
'put off' becomes 'defer' (synonyms).
'losing heart' suggests removing the centre.
'defer' with its centre taken out is 'DEER'.
(Other definitions for deer that I've seen before include "Like a power switch" , "Wild animal" , "Grazing animal(s)" , "Bucks and does?" , "Hoofed animals" .)