One runs twice with it, to worker's annoyance (8)
I believe the answer is:
irritant
'annoyance' is the definition.
(irritant is a kind of annoyance)
'one runs twice with it to worker's' is the wordplay.
'one' becomes 'i' (Roman numeral).
'runs' becomes 'r' (cricket abbreviation**).
'twice' indicates the doubling of some letters.
'with' means one lot of letters go next to another.
'to worker' becomes 'ant' (worker is a type of ant).
'r' duplicated is 'rr'.
'i'+'rr'+'it'+'ant'='IRRITANT'
(Other definitions for irritant that I've seen before include "distraction" , "Annoying thing" , "Annoying factor" , "causing annoyance" , "Something that chafes or annoys" .)