Dad with delayed sense of taste (6)
I believe the answer is:
palate
'sense of taste' is the definition.
(palate can mean someone's ability to taste)
'dad with delayed' is the wordplay.
'dad' becomes 'pa' (both can mean one's father).
'with' says to put letters next to each other.
'delayed' becomes 'late' (running late).
'pa'+'late'='PALATE'
(Other definitions for palate that I've seen before include "A plate (anag.)" , "relish" , "One may savour" , "Sense of taste at roof of mouth" , "having ability to appreciate good food" .)