Cut chap short and smile cheekily, causing indignation (7)
I believe the answer is:
chagrin
'indignation' is the definition.
'cut chap short and' is the wordplay.
'cut' means to remove the last letter.
'short and' becomes 'grin' (I am not sure about this - if you are sure you should believe this answer much more).
'chap' with its last letter removed is 'cha'.
'cha'+'grin'='CHAGRIN'
'smile cheekily causing' is the link.
This may not be correct. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for chagrin that I've seen before include "Annoy" , "Frustration at having failed" , "Annoyance/shame at one's failure" , "Humiliation, mortification" , "Feeling of annoyance and embarrassment" .)