Giant cuts quashed by extremely canny Conservative (7)
I believe the answer is:
cyclops
'giant' is the definition.
(one-eyed giant in Greek myth)
'cuts quashed by extremely canny conservative' is the wordplay.
'cuts' becomes 'lops' (lopping is a kind of cutting).
'quashed by' is a charade indicator (letters next to each other) (some letters go on top of or 'quash' others).
'extremely' suggests removing the centre (only the extremes of the word are used).
'conservative' becomes 'C' (UK political abbreviation).
'canny' with its centre removed is 'cy'.
'cy'+'c'='cyc'
'lops' put after 'cyc' is 'CYCLOPS'.
(Other definitions for cyclops that I've seen before include "Fabled one-eyed giant" , "Electronic tennis aid" , "The mythical giant with a single eye in the middle of his forehead" , "A one-eyed giant in Greek myth" , "In Greek myth, one of a race of one-eyed giants" .)