It's worse for a mouse than for a greyhound (4)
I believe the answer is:
trap
'a mouse than for a greyhound' is the definition.
I can't judge whether this defines the answer.
'it's worse' is the wordplay.
'it' becomes ''t' (abbreviation. e.g. in 'tis).
'worse' becomes 'rap' (I can't justify this - if you can you should give a lot more credence to this answer).
't'+'rap'='TRAP'
'for' acts as a link.
Can you help me to learn more?
(Other definitions for trap that I've seen before include "that may be sprung?" , "that can let an actor down" , "fault" , "Light carriage; ensnare" , "Snaring device" .)