Annoyance with old Italian sauce (5)
I believe the answer is:
pesto
'italian sauce' is the definition.
(I've seen this before)
'annoyance with old' is the wordplay.
'annoyance' becomes 'pest' (I've seen this before).
'with' says to put letters next to each other.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'pest'+'o'='PESTO'
(Other definitions for pesto that I've seen before include "A sauce served with pasta" , "Sauce made of basil and cheese" , "Green sauce" , "Basic sauce" , "Basil-based sauce" .)