Greek breaking link displays annoyance (7)
I believe the answer is:
'annoyance' is the definition.
(I've seen this before)
'greek breaking link' is the wordplay.
'greek' becomes 'gr' (abbreviation for Greek).
'breaking' indicates putting letters inside.
'link' becomes 'chain' (I can't justify this - if you can you should believe this answer much more).
'gr' going within 'chain' is 'CHAGRIN'.
'displays' is the link.
Can you help me to learn more?
(Other definitions for chagrin that I've seen before include "Discomfiture" , "Feeling of vexation or embarrassment" , "irritation?" , "Disappointment, exasperation" , "Shame at one's failure" .)