Mayflower passenger turning cheek, resolute (7)
I believe the answer is:
pilgrim
'mayflower passenger' is the definition.
(I've seen this in another clue)
'turning cheek resolute' is the wordplay.
'turning' shows that the letters should be reversed in order.
'cheek' becomes 'lip' (both can mean insolence or rudeness).
'resolute' becomes 'grim' (I've seen this before).
'lip' in reverse letter order is 'pil'.
'pil'+'grim'='PILGRIM'
(Other definitions for pilgrim that I've seen before include "Wanderer" , "One of those who sailed to New England in the Mayflower, 1620" , "Traveller in sacred place for religious reasons" , "Traveller to a sacred place like Mecca" , "Wayfarer to holy site" .)