Men risk losing their heads to improve (7)
I believe the answer is:
enhance
'to improve' is the definition.
(enhancing is a kind of improving)
'men risk losing their heads' is the wordplay.
'men' becomes 'en' (I can't justify this - if you can you should believe this answer much more).
'risk' becomes 'chance' (synonyms).
'losing their heads' means to remove the first letter.
'chance' with its initial letter taken off is 'hance'.
'en'+'hance'='ENHANCE'
Can you help me to learn more?
(Other definitions for enhance that I've seen before include "Make better or more attractive" , "lift" , "Improve the quality of (something)" , "Improve quality (of item)" , "Add function, appeal, to" .)