Modest or awfully rude, me? (6)
I believe the answer is:
demure
'modest' is the definition.
(I've seen this in another clue)
'awfully rude me?' is the wordplay.
'awfully' is an anagram indicator.
'rude'+'me'='rudeme'
'rudeme' with letters rearranged gives 'DEMURE'.
'or' is the link.
(Other definitions for demure that I've seen before include "Modestly behaved" , "Coy, shy" , "Affectedly quiet and serious" , "Prim, coy" , "Prim and proper" .)