Flush and terribly sorry, having lost heart (4)
I believe the answer is:
rosy
'flush' is the definition.
(both can mean red in the face)
'terribly sorry having lost heart' is the wordplay.
'terribly' indicates an anagram.
'having lost heart' means to remove the middle letters.
'sorry' with its middle taken out is 'sory'.
'sory' anagrammed gives 'ROSY'.
'and' acts as a link.
(Other definitions for rosy that I've seen before include "Blooming - hopeful" , "lady" , "Hopeful; blushing" , "Of the future, promising like flower" , "Healthy-looking" .)