Frenchman Chechen rightly includes (5)
I believe the answer is:
henri
'frenchman' is the definition.
(Henri matches up with the definition frenchman)
'chechen rightly includes' is the wordplay.
'includes' says the answer is hidden in the clue.
'HENRI' is hidden in the letters of 'chechen rightly'.
(Other definitions for henri that I've seen before include "British poet" , "-- Matisse, French artist" , "French name" .)