Short game in which to see the French shine (5)
I believe the answer is:
gleam
'shine' is the definition.
(gleam is a kind of shine)
'short game in which to see the french' is the wordplay.
'short' means to remove the last letter.
'in which to see' indicates putting letters inside.
'the french' becomes 'le'.
'game' with its last letter taken off is 'gam'.
'gam' placed around 'le' is 'GLEAM'.
(Other definitions for gleam that I've seen before include "Glimmer, lustre" , "Sparkle, glimmer" , "Brief show of soft light" , "Glisten" , "Beam of soft light, comes and goes" .)