Animation of mouse and fly involving line in unctuous style (9)
I believe the answer is:
fulsomely
'in unctuous style' is the definition.
(fulsome can mean flattering or unctuous)
'animation of mouse and fly involving line' is the wordplay.
'animation of' indicates an anagram (letters animated or moving ).
'and' is a charade indicator (letters next to each other).
'involving' is an insertion indicator.
'line' becomes 'l' (used when specifying particular lines from a poem).
'mouse' after 'fly' is 'flymouse'.
'flymouse' anagrammed gives 'fusomely'.
'fusomely' placed around 'l' is 'FULSOMELY'.
(Other definitions for fulsomely that I've seen before include "In an unctuous, insincere manner" , "displaying sickening obsequiousness" .)