Case cracked: 500 held by French city authority (10)
I believe the answer is:
ascendancy
'authority' is the definition.
(thesaurus)
'case cracked 500 held by french city' is the wordplay.
'cracked' indicates an anagram.
'500' becomes 'D' (Roman numeral for 500).
'held by' indicates putting letters inside.
'french city' becomes 'Nancy' (city in France).
'd' going into 'nancy' is 'ndancy'.
'case' anagrammed gives 'asce'.
'asce'+'ndancy'='ASCENDANCY'
(Other definitions for ascendancy that I've seen before include "Condition of being dominant" , "Dominating influence (A, not E)" , "Influential position" , "sway" , "Dominance" .)