Prompt to accept one's at home cooking (7)
I believe the answer is:
cuisine
'cooking' is the definition.
('cuisine' can be a synonym of 'cooking')
'prompt to accept one's at home' is the wordplay.
'prompt' becomes 'cue' ('cue' can be a synonym of 'prompt').
'to accept' indicates putting letters inside.
'one's' becomes 'is' (Roman numeral).
'at home' becomes 'in' ('in' can be similar in meaning to 'at home').
'is'+'in'='isin'
'cue' going around 'isin' is 'CUISINE'.
(Other definitions for cuisine that I've seen before include "Culinary art" , "The food" , "Cooking style characteristic of a place" , "Haute . . . . . . . is high-class cooking" , "Local cooking style" .)