Begging access, endlessly worrying (10)
I believe the answer is:
entreating
'begging' is the definition.
(I've seen this in another clue)
'access endlessly worrying' is the wordplay.
'access' becomes 'entry' (entry is a kind of access).
'endlessly' means to remove the last letter.
'worrying' becomes 'eating' (eating is a kind of worrying).
'entry' with its last letter taken away is 'entr'.
'entr'+'eating'='ENTREATING'
(Other definitions for entreating that I've seen before include "Imploring" , "Beseeching" , "Asking (someone) earnestly" .)