A private person? (5)
I believe the answer is:
tommy
'a private' is the definition.
(thesaurus)
'person?' is the wordplay.
I cannot really understand how this works, but
'person?' could be 'tom' (Tom is an example) and 'tom' is found in the answer.
The remaining letters 'my' is a valid word which might be clued in a way I don't see.
This explanation may well be incorrect...
(Other definitions for tommy that I've seen before include "British private soldier (colloq.)" , "British soldier (informal)" , "Bread" , "Private in the Army - kind of gun" , "Private soldier (slang)" .)