Greek character's teacup regularly avoided (3)
I believe the answer is:
tau
'greek character's' is the definition.
(letter from the Greek alphabet)
'teacup regularly avoided' is the wordplay.
'regularly avoided' means one should take alternating letters.
The alternating letters of 'teacup' are 'TAU'.
(Other definitions for tau that I've seen before include "character from Athens" , "letter from foreigner" , "19th letter of the Greek alphabet" , "Letter from Greece" , "Foreign character" .)