Eat in the empty French Head for a private conversation (4-1-4)
I believe the answer is:
tete-a-tete
'private conversation' is the definition.
(I've seen this before)
'eat in the empty french head' is the wordplay.
'in' indicates putting letters inside.
'empty' says to hollow out the word (remove centre letters).
'french head' becomes 'tete' ('head' in French).
'the' with its centre removed is 'te'.
'te'+'tete'='tetete'
'eat' placed inside 'tetete' is 'TETE-A-TETE'.
'for a' is the link.
(Other definitions for tete-a-tete that I've seen before include "Private couple's conversation" , "One-to-one meeting" , "Informal chat" , "Private conversation between two French people" , "Private meeting" .)