Sweetheart having fling -- I'd err terribly (10)
I believe the answer is:
girlfriend
'sweetheart' is the definition.
(I've seen this before)
'fling i'd err terribly' is the wordplay.
'terribly' is an anagram indicator.
'fling'+'id'+'err'='flingiderr'
'flingiderr' anagrammed gives 'GIRLFRIEND'.
'having' is the link.
(Other definitions for girlfriend that I've seen before include "Female squeeze" , "Sweetheart" , "Female companion" , "date" .)