Ring after annoying person gives one sauce (5)
I believe the answer is:
pesto
'one sauce' is the definition.
(I've seen this before)
'ring after annoying person' is the wordplay.
'ring' becomes 'o'.
'after' means one lot of letters go next to another.
'annoying person' becomes 'pest' (I've seen this before).
'o' put after 'pest' is 'PESTO'.
'gives' acts as a link.
(Other definitions for pesto that I've seen before include "A sauce served with pasta" , "Ligurian food" , "Italian sauce containing basil" , "Basil-based sauce" , "Pasta sauce variety" .)