Old kingdom beginning to adapt, with thanks to the French (6)
I believe the answer is:
mercia
'old kingdom' is the definition.
(Anglo-Saxon kingdom)
'beginning to adapt with thanks to the french' is the wordplay.
'beginning to' suggests taking the first letters.
'with' says to put letters next to each other.
'thanks to the french' becomes 'merci' ('thank you' in French).
The initial letter of 'adapt' is 'a'.
'a' after 'merci' is 'MERCIA'.
(Other definitions for mercia that I've seen before include "part of old England" , "Former kingdom centred on Tamworth" , "Ancient British kingdom" , "Ancient region" , "Ancient midland kingdom" .)