Put off animal, with time (5)
I believe the answer is:
deter
'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' means one lot of letters goes inside another.
'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" .)