Achievement by that woman is something insignificant (7)
I believe the answer is:
feather
'something insignificant' is the definition.
(feather can mean something unimportant)
'achievement by that woman' is the wordplay.
'achievement' becomes 'feat' (feat is a kind of achievement).
'by' is a charade indicator (letters next to each other).
'that woman' becomes 'her' (her refers to a woman).
'feat'+'her'='FEATHER'
'is' acts as a link.
(Other definitions for feather that I've seen before include "Part of a bird's plumage" , "English plume (de ma tante?)" , "Free hat (anag)" , "Prov. light part of bird" , "Rotate (propeller) blades (to reduce thrust)" .)