Refinement of French priest going around last month (7)
I believe the answer is:
culture
'refinement' is the definition.
(I know that refinement can be written as culture)
'french priest going around last month' is the wordplay.
'french priest' becomes 'cure' (French for 'priest').
'going around' means one lot of letters goes inside another.
'last month' becomes 'ult' (short for 'last month' in correspondence).
'cure' placed around 'ult' is 'CULTURE'.
'of' is the link.
(Other definitions for culture that I've seen before include "Grown preparation of cells; the arts" , "Disorientation caused by a complete change environment" , "Customs of a group" , "Set of ideas and values of a group" , "Grow (eg, bacteria, tissue cells) in artificial media" .)