Cross after the French becoming sloppy (3)
I believe the answer is:
lax
'sloppy' is the definition.
(I've seen this before)
'cross after the french' is the wordplay.
'cross' becomes 'x' (X is cross-shaped).
'after' means one lot of letters go next to another.
'the french' becomes 'la' ('the' in French).
'x' after 'la' is 'LAX'.
'becoming' acts as a link.
(Other definitions for lax that I've seen before include "Remiss" , "Flaccid" , "Slack, loose" , "Not sufficiently strict" , "Insufficiently strict" .)