Bride must ignore free thank you letter from abroad (4)
I believe the answer is:
beta
'letter from abroad' is the definition.
(letter of the Greek alphabet)
'bride must ignore free thank you' is the wordplay.
'must ignore' suggests deleting specific letters.
'free' becomes 'rid' (similar in meaning).
'thank you' becomes 'ta'.
'bride' with 'rid' taken away is 'be'.
'be'+'ta'='BETA'
(Other definitions for beta that I've seen before include "Second Greek letter - beat that" , "Greek letter -- second in order of importance" , "Second letter of the Greek alphabet" , "character in Marathon" , "Second-rate" .)