The garment with which a woman may put on airs? (4)
I believe the answer is:
sari
'the garment' is the definition.
(sari is a kind of garment)
'which a woman may put on airs?' is the wordplay.
'airs' anagrams to 'SARI'.
But, I'm unsure how the anagram is indicated.
'with' is the link.
Can you help me to learn more?
(Other definitions for sari that I've seen before include "Hindu female garment" , "South Asian garment (variant spelling)" , "Traditional Indian women's dress" , "Eastern garment" , "Hindu woman's dress" .)