Animal put off, losing heart (4)
I believe the answer is:
deer
'animal' is the definition.
(thesaurus)
'put off losing heart' is the wordplay.
'put off' becomes 'defer' ('defer' can be a synonym of 'put off').
'losing heart' suggests removing the centre.
'defer' with its centre taken out is 'DEER'.
(Other definitions for deer that I've seen before include "Sledge animals for Father Christmas" , "The - Hunter, 1978 film" , "Perhaps fallow" , "does for example" , "Bucks and does?" .)