Eg Brendan finishes off tricky puzzle that's irritating to face (4)
I believe the answer is:
stye
'irritating to face' is the definition.
(inflamed swelling on eyelid)
'eg brendan finishes off tricky puzzle' is the wordplay.
'eg brendan' becomes 'St' (short for saint: one of 12 apostles of Ireland).
'finishes off' says to take the final letters (take the finishing letters off of the words).
The last letters of 'tricky puzzle' are 'ye'.
'st'+'ye'='STYE'
'that's' is the link.
(Other definitions for stye that I've seen before include "nasty spot" , "irritation to the viewer" , "Inflamed swelling on the edge of an eyelid" , "Eyelid problem" , "Eyelid affliction" .)