See about skinning cats; approach gradually (6,2)
I believe the answer is:
attend to
'see about' is the definition.
(synonyms)
'skinning cats approach gradually' is the wordplay.
'skinning' says to take the centre (the word is skinned of its outside letters).
'approach gradually' becomes 'tend to' (tend to can mean to approach or tend towards).
The middle letters of 'cats' are 'at'.
'at'+'tendto'='ATTEND TO'
(Other definitions for attend to that I've seen before include "Deal with" , "Provide for the needs of" , "Take seriously - wait on" .)