... an offence that makes French very cross (8)
I believe the answer is:
trespass
'an offence' is the definition.
(I've seen this before)
'french very cross' is the wordplay.
'french very' becomes 'tres' ('very' in French).
'cross' becomes 'pass'.
'tres'+'pass'='TRESPASS'
'that makes' is the link.
(Other definitions for trespass that I've seen before include "Go out of bounds" , "Enter another's property unlawfully" , "Transgress" , "Enter land illegally" , "Enter someone else's property illegally" .)