Count of French prison beginning to despair (6)
I believe the answer is:
depend
'count' is the definition.
('count on me' and 'depend on me' are synonymous)
'of french prison beginning to despair' is the wordplay.
'of french' becomes 'de'.
'prison' becomes 'pen' (I've seen this before).
'beginning to' indicates taking the first letters.
The initial letter of 'despair' is 'd'.
'de'+'pen'+'d'='DEPEND'
(Other definitions for depend that I've seen before include "Be unable to do without, rely" , "Count on or hang from" , "Rely (upon)" , "Rely or hang on" , "Dangle - rely" .)