Time with former lovers consuming Frenchman’s cuisine (3-3)
I believe the answer is:
tex-mex
'cuisine' is the definition.
'time with former lovers consuming frenchman's' is the wordplay.
'time' becomes 't'.
'with' means one lot of letters go next to another.
'former lovers' becomes 'exex' (former lover is an ex).
'consuming' is an insertion indicator.
'frenchman' becomes 'm' (abbreviation for Monsieur - like 'Mr').
'exex' enclosing 'm' is 'exmex'.
't'+'exmex'='TEX-MEX'
(Other definitions for tex-mex that I've seen before include "Cuisine of southern North America" , "N. American cuisine" , "North-American cuisine" , "Style of food" .)