After rejection, half-hearted offspring proceeded cautiously (5)
I believe the answer is:
nosed
'proceeded cautiously' is the definition.
(to nose along is to move along slowly)
'after rejection half-hearted offspring' is the wordplay.
'after rejection half' becomes 'nose' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'hearted' means to look at the middle letters.
'offspring' becomes 'foundling' (foundling is a kind of offspring).
The centre of 'foundling' is 'd'.
'nose'+'d'='NOSED'
(Other definitions for nosed that I've seen before include "Checked wine's aroma" , "proceeded gingerly" , "Move back or away" , "Moved slowly and cautiously forward" , "Started judging wine, say" .)