Half hidden in a private place (3)
I believe the answer is:
den
'a private place' is the definition.
'half hidden' is the wordplay.
'half' suggests halving the letters.
'hidden' cut in half is 'DEN'.
'in' is the link.
(Other definitions for den that I've seen before include "Wild animal's home" , "Private office" , "Animal's home or gent's study" , "Cave" , "Lair for animal or study for gent" .)