One giving instruction thus regularly rejected rubbish (5)
I believe the answer is:
tutor
'one giving instruction' is the definition.
(a tutor instructs pupils)
'thus regularly rejected rubbish' is the wordplay.
'regularly' means one should take alternating letters (regularly take one letter, leave next etc.).
'rejected' shows that the letters should be reversed in order.
'rubbish' becomes 'rot' (both can mean nonsense).
The alternating letters of 'thus' are 'tu'.
'rot' written backwards gives 'tor'.
'tu'+'tor'='TUTOR'
(Other definitions for tutor that I've seen before include "Private or university teacher" , "Private instructor" , "Individual teacher" , "Master" , "college teacher" .)