We will be not half embarrassed and polite (4-4)
I believe the answer is:
well-bred
'polite' is the definition.
(brought up well)
'we will be not half embarrassed' is the wordplay.
'will' becomes ''ll' (contracted form, as in 'he'll' - 'he will').
'not half' indicates taking half.
'embarrassed' becomes 'red' (someone may blush or go red when embarrassed).
'be' cut in half is 'b'.
'we'+'ll'+'b'+'red'='WELL-BRED'
'and' is the link.
(Other definitions for well-bred that I've seen before include "Properly brought up" , "Of good stock or having good manners ( 4-4)" , "Educated in good manners" .)