Detective quiet about the French unit losing heart (6)
I believe the answer is:
sleuth
'detective' is the definition.
(I know that sleuth is a type of detective)
'quiet about the french unit losing heart' is the wordplay.
'quiet' becomes 'sh' (both can be said to tell someone to be quiet).
'about' means one lot of letters goes inside another.
'the french' becomes 'le' ('the' in French).
'losing heart' suggests removing the centre.
'unit' with its middle taken out is 'ut'.
'le'+'ut'='leut'
'sh' going around 'leut' is 'SLEUTH'.
(Other definitions for sleuth that I've seen before include "A detective like Sherlock - given to hustle" , "Investigator (informal)" , "Follow" , "Detective; play and film" , "Holmes, for one" .)