Cautious about egghead being tired (5)
I believe the answer is:
weary
'being tired' is the definition.
('weary' can be similar in meaning to 'tired')
'cautious about egghead' is the wordplay.
'cautious' becomes 'wary' (similar in meaning).
'about' is an insertion indicator.
'egghead' becomes 'e' (head letter of 'egg').
'wary' going around 'e' is 'WEARY'.
(Other definitions for weary that I've seen before include "Worn-out" , "Flagging" , "Enervated" , "spent" , "Physically and mentally fatigued" .)