Good to beat up bloodsucker (4)
I believe the answer is:
gnat
'bloodsucker' is the definition.
(I've seen this before)
'good to beat up' is the wordplay.
'good' becomes 'g' (abbreviation).
'to beat' becomes 'tan' (I've seen this before).
'up' says the letters should be written in reverse (in down clue: letters go upwards).
'tan' in reverse letter order is 'nat'.
'g'+'nat'='GNAT'
(Other definitions for gnat that I've seen before include "annoying flier" , "Midge, mosquito" , "Small (swarming) fly" , "little irritation" , "a little bit of a nuisance" .)