Fine distinction going round is something that annoys us (8)
I believe the answer is:
nuisance
'something that annoys us' is the definition.
(a nuisance is something annoying or inconvenient)
'fine distinction going round is' is the wordplay.
'fine distinction' becomes 'nuance' (a nuance is a precise distinction).
'going round' indicates putting letters inside (some letters go around others).
'nuance' going around 'is' is 'NUISANCE'.
(Other definitions for nuisance that I've seen before include "It causes inconvenience or annoyance" , "Thing causing annoyance" , "Source of irritation, person or thing" , "Annoying or troublesome person or thing" , "Pest or irritation" .)