Job of Tut, primarily, when king (4)
I believe the answer is:
task
'job' is the definition.
('task' can be a synonym of 'job')
'tut primarily when king' is the wordplay.
'primarily' suggests taking the first letters.
'when' becomes 'as'.
'king' becomes 'k'.
The initial letter of 'tut' is 't'.
't'+'as'+'k'='TASK'
'of' is the link.
(Other definitions for task that I've seen before include "Give mission" , "Allocate work" , "Assignment" , "Job to be done" , "Chore, job." .)