Duke having private meal (6)
I believe the answer is:
dinner
'meal' is the definition.
(I know that dinner is a type of meal)
'duke having private' is the wordplay.
'duke' becomes 'd' (abbreviation).
'having' means one lot of letters go next to another.
'private' becomes 'inner' (as in one's private or inner thoughts).
'd'+'inner'='DINNER'
(Other definitions for dinner that I've seen before include "Evening meal" , "Supper" , "one making a row?" , "grub" , "Eat it in the strange red inn" .)