Overcome impudence that holds state up (7)
I believe the answer is:
prevail
'overcome' is the definition.
(thesaurus)
'impudence that holds state up' is the wordplay.
'impudence' becomes 'lip' (lip is a kind of impudence).
'that holds' is an insertion indicator.
'state' becomes 'aver' (both can mean to declare).
'up' says the letters should be written backwards (in down clue: letters go upwards).
'lip' going around 'aver' is 'liaverp'.
'liaverp' back-to-front is 'PREVAIL'.
(Other definitions for prevail that I've seen before include "Prove more powerful" , "Carry the day" , "get the upper hand" , "Get one's way" , "Have opinion accepted" .)