Case of 'Physician, heal thyself'? That's a bore (5)
I believe the answer is:
drill
'a bore' is the definition.
(bore can mean to drill into something)
'case of 'physician heal thyself'?' is the wordplay.
I cannot really understand how this works, but
''physician' could be 'dr' (synonyms) and 'dr' is located in the answer.
The remaining letters 'ill' is a valid word which might be clued in a way I don't understand.
This explanation may well be incorrect...
'that's' is the link.
(Other definitions for drill that I've seen before include "Hole-making tool" , "makes one exercise" , "School" , "(Correct) procedure" , "Monkey" .)