After tea laugh at one’s discomfiture (7)
I believe the answer is:
chagrin
'one's discomfiture' is the definition.
'chagrin' can be an answer for 'discomfiture' (thesaurus). I am not certain of the 'one's' bit.
'after tea laugh' is the wordplay.
'after' says to put letters next to each other.
'tea' becomes 'cha' (cha is a type of tea).
'laugh' becomes 'grin' (I can't justify this - if you can you should give a lot more credence to this answer).
'cha'+'grin' is 'CHAGRIN'.
'at' acts as a link.
Can you help me to learn more?
(Other definitions for chagrin that I've seen before include "Shame at having failed" , "Embarrassment - rich nag (anag)" , "Humiliation, mortification" , "Disappointment, exasperation" , "Arching with annoyed disappointment" .)