Problem involving upper-class person acting insincerely (6)
I believe the answer is:
poseur
'person acting insincerely' is the definition.
'problem involving upper-class' is the wordplay.
'problem' becomes 'poser' (poser is a kind of problem).
'involving' means one lot of letters goes inside another.
'upper-class' becomes 'u' ('u' can be similar in meaning to 'upper-class').
'poser' enclosing 'u' is 'POSEUR'.
(Other definitions for poseur that I've seen before include "Pseud, posturer" , "One seeking to impress others with affected style" , "Pretentious chap" , "One who strikes an attitude to impress" , "Affected person, strikes attitudes" .)