Hare grabbing a rodent, scratching tail very well (8)
I believe the answer is:
famously
'very well' is the definition.
(thesaurus)
'hare grabbing a rodent scratching tail' is the wordplay.
'hare' becomes 'fly' (both can mean to run fast).
'grabbing' indicates putting letters inside.
'rodent' becomes 'mouse' (mouse is a kind of rodent).
'scratching tail' means to remove the last letter.
'a'+'mouse'='amouse'
'amouse' with its last letter taken away is 'amous'.
'fly' placed around 'amous' is 'FAMOUSLY'.
(Another definition for famously that I've seen is " Well - in a way that's well known".)