Time to take heart from award (4)
I believe the answer is:
hour
'time' is the definition.
(hour is a kind of time)
'to take heart from award' is the wordplay.
'to take heart from' suggests removing the centre.
'award' becomes 'honour' ('honour' can be a synonym of 'award').
'honour' with its middle taken out is 'HOUR'.
(Other definitions for hour that I've seen before include "Time within day" , "Part of day" , "(Period of) time" , "Period of day" , "Three thousand and six hundred seconds" .)