Sulks about test woman gives relative (10)
I believe the answer is:
stepmother
'relative' is the definition.
(stepmother is a kind of relative)
'sulks about test woman' is the wordplay.
'sulks' becomes 'pets' (pet can mean a sulk or tantrum).
'about' says the letters should be written in reverse.
'test' becomes 'MOT' (as in an MOT test for a vehicle).
'woman' becomes 'her' (pronoun for a woman).
'pets' in reverse letter order is 'step'.
'step'+'mot'+'her'='STEPMOTHER'
'gives' is the link.
(Other definitions for stepmother that I've seen before include "A relation by marriage" , "Stereotypically wicked member of 3" , "Relative by a further marriage of one's father" , "not a natural parent" , "Wicked character of fairy stories" .)