Greeks initially beat back bloodsucker (4)
I believe the answer is:
gnat
'bloodsucker' is the definition.
(I've seen this before)
'greeks initially beat back' is the wordplay.
'initially' suggests taking the first letters.
'beat' becomes 'tan' (tan can mean to beat or hit).
'back' says the letters should be written backwards.
The initial letter of 'greeks' is 'g'.
'tan' in reverse letter order is 'nat'.
'g'+'nat'='GNAT'
(Other definitions for gnat that I've seen before include "Midge, mosquito" , "Insect often hovering over stagnant water" , "Insect like small mosquito" , "a little bit of a nuisance" , "Small biting flying insect" .)