Strong at last beat up bloodsucker (4)
I believe the answer is:
gnat
'bloodsucker' is the definition.
(I've seen this before)
'strong at last beat up' is the wordplay.
'at last' says to take the final letters.
'beat' becomes 'tan' (tan can mean to beat or hit).
'up' is a reversal indicator (in down clue: letters go upwards).
The last letter of 'strong' is 'g'.
'tan' reversed gives 'nat'.
'g'+'nat'='GNAT'
(Other definitions for gnat that I've seen before include "Woodland area" , "Tang (anag.)" , "Sucker (if female)" , "Small bloodsucker" , "mozzie" .)