Relative is haggard — not good! (4)
I believe the answer is:
aunt
'relative' is the definition.
(aunt is a kind of relative)
'haggard not good' is the wordplay.
'haggard' becomes 'gaunt' (synonyms).
'not' is a deletion indicator.
'good' becomes 'g' (abbreviation).
'gaunt' with 'g' taken out is 'AUNT'.
'is' acts as a link.
(Other definitions for aunt that I've seen before include "Tuna (anag.)" , "Uncle's wife" , "Female relation" , "Parent's sister" , "Wife of one's uncle" .)