Private's pained expression, close to exhaustion (3)
I believe the answer is:
own
'private's' is the definition.
(as in one's own or private belongings)
'pained expression close to exhaustion' is the wordplay.
'pained expression' becomes 'ow' (exclamation of pain).
'close to' says to take the final letters (the 'close' to the word).
The final letter of 'exhaustion' is 'n'.
'ow'+'n'='OWN'
(Other definitions for own that I've seen before include "Acknowledge" , "Possess or confess" , "Be in legal possession of (3)" , "Posses" , "Personal; possess" .)