Eatery set out tart to air (9)
I believe the answer is:
trattoria
'eatery' is the definition.
(I know that Trattoria is a type of eatery)
'set out tart to air' is the wordplay.
'set out' is an anagram indicator.
'tart'+'to'+'air'='tarttoair'
'tarttoair' anagrammed gives 'TRATTORIA'.
(Other definitions for trattoria that I've seen before include "Italian eating establishment" , "Italian restaurant" , "Place to eat" , "where Italians eat" .)