Put off animal with time (5)
I believe the answer is:
'put off' is the definition.
(both can mean to discourage)
'animal with time' is the wordplay.
'animal' becomes 'deer' (deer is a type of animal).
'with' indicates putting letters inside.
'time' becomes 't' (abbreviation).
'deer' enclosing 't' is 'DETER'.
(Other definitions for deter that I've seen before include "Warn off" , "stop" , "Frighten off; prevent" , "Inhibit" , "Discourage or impede" .)