Handle short cut holding chap up (8)
I believe the answer is:
nickname
'handle' is the definition.
(a handle is a name)
'cut holding chap up' is the wordplay.
'cut' becomes 'nick' (nicking is a kind of cutting).
'holding chap up' becomes 'name' (I can't explain this - if you can you should give a lot more credence to this answer).
'nick'+'name'='NICKNAME'
'short' acts as a link.
Can you help me to learn more?
(Other definitions for nickname that I've seen before include "Sobriquet" , "Jack the Ripper, for example" , "Pet appellation" , "Affectionate or invented appellation, not one's own" , "Cognomen" .)