One’s going into fine distinction — bother! (8)
I believe the answer is:
nuisance
'bother' is the definition.
(I know that nuisance is a type of bother)
'one's going into fine distinction' is the wordplay.
'one's' becomes 'I's' (Roman numeral).
'going into' means one lot of letters goes inside another.
'fine distinction' becomes 'nuance' (a nuance is a precise distinction).
'is' inserted within 'nuance' is 'NUISANCE'.
(Other definitions for nuisance that I've seen before include "Bothersome pest" , "Something which annoys" , "Pest, source of annoyance" , "Irritant or botheration" , "See 16" .)