Staff argue regularly and run for supervisor (7)
I believe the answer is:
manager
'supervisor' is the definition.
(thesaurus)
'staff argue regularly and run' is the wordplay.
'staff' becomes 'man' (manning is a kind of staffing).
'regularly' indicates alternate letters (regularly take one letter, leave next etc.).
'and' means one lot of letters go next to another.
'run' becomes 'r' (cricket abbreviation).
The alternating letters of 'argue' are 'age'.
'man'+'age'+'r'='MANAGER'
'for' is the link.
(Other definitions for manager that I've seen before include "who sits on the bench?" , "Someone in control (or who copes?)" , "suit?" , "One in charge of sports team" , "Sports team controller" .)