This relation’s gibe, not the first (4)
I believe the answer is:
aunt
'this relation's' is the definition.
(aunt is a kind of relation)
'gibe not the first' is the wordplay.
'gibe' becomes 'taunt' (I've seen this before).
'not the first' means to remove the first letter.
'taunt' with its initial letter taken off is 'AUNT'.
(Other definitions for aunt that I've seen before include "Parent's sister" , "Charley's ___ , farce staged in London in 1892, revived and adapted ever since" , "Put up with" , "Charley's in play?" , "Sister of father/mother" .)