Delicacy isn't commonly penetrating outskirts of Dudley (6)
I believe the answer is:
dainty
'delicacy' is the definition.
('dainty' can be a synonym of 'delicacy')
'isn't commonly penetrating outskirts of dudley' is the wordplay.
'isn't commonly' becomes 'aint' (I've seen this in other clues).
'penetrating' is an insertion indicator.
'outskirts of' says to hollow out the word (remove centre letters).
'dudley' with its centre taken out is 'dy'.
'aint' inserted inside 'dy' is 'DAINTY'.
(Other definitions for dainty that I've seen before include "Refined" , "Delicate, tasteful" , "Exquisite" , "Small and beautiful" , "Piece of cake!" .)