Influence criminal case due to begin in French city (10)
I believe the answer is:
ascendancy
'influence' is the definition.
(I have seen 'Dominating influence ' mean 'ascendancy' so perhaps 'influence' could also mean 'ascendancy')
'criminal case due to begin in french city' is the wordplay.
'criminal' indicates anagramming the letters.
'to begin' says to take the initial letters.
'in' is an insertion indicator.
'french city' becomes 'nancy' (I've seen this before**).
The first letter of 'due' is 'd'.
'd' placed within 'nancy' is 'ndancy'.
'case' is an anagram of 'asce'.
'asce'+'ndancy'='ASCENDANCY'
(Other definitions for ascendancy that I've seen before include "Dominance" , "Candy canes (anag.)" , "Domination" , "sway" , "upper hand" .)