Witty style in a new essay following requests (10)
I believe the answer is:
pleasantry
'witty style' is the definition.
'a new essay following requests' is the wordplay.
'a new' becomes 'an' (I've seen this in other clues).
'essay' becomes 'try' ('try' can be a synonym of 'essay').
'following' means one lot of letters go next to another.
'requests' becomes 'pleas' (plea is a kind of request**).
'an'+'try'='antry'
'antry' after 'pleas' is 'PLEASANTRY'.
'in' acts as a link.
(Other definitions for pleasantry that I've seen before include "Say, how are you doing" , "Jocular comment" , "Polite remark" , "jocularity" , "facetious trick" .)