That part of the skin cut, I see, by the French (7)
I believe the answer is:
cuticle
'that part of the skin' is the definition.
('cuticle' is part of 'skin')
'cut i see by the french' is the wordplay.
'see' becomes 'c' (the word for the letter, according to Chambers).
'by' is a charade indicator (letters next to each other).
'the french' becomes 'le' (**).
'cut'+'i'+'c'+'le'='CUTICLE'
(Other definitions for cuticle that I've seen before include "Skin at foot of nail" , "layer of protection" , "Protective layer" , "Skin around the nail" , "Dead skin on hand" .)