Oddly this sign of embarrassment reveals fatigue (9)
I believe the answer is:
tiredness
'fatigue' is the definition.
('tiredness' can be a synonym of 'fatigue')
'oddly this sign of embarrassment' is the wordplay.
'oddly' means one should take alternating letters (letters in the odd-numbered positions are used).
'sign of embarrassment' becomes 'redness'.
The alternating letters of 'this' are 'ti'.
'ti'+'redness'='TIREDNESS'
'reveals' acts as a link.
(Other definitions for tiredness that I've seen before include "Exhaustion" , "Residents in state of fatigue" , "That's liable to occur at midnight" , "Weary feeling" , "Lassitude; residents (anag.)" .)