Becoming disheartened, notice another birthday is not far off (6)
I believe the answer is:
nearby
'not far off' is the definition.
(I've seen this before)
'becoming disheartened notice another birthday' is the wordplay.
I cannot really understand how this works, but
'notice' could be 'ear' (I've seen this in other clues) and 'ear' is found in the answer.
This explanation may well be incorrect...
'is' acts as a link.
Can you help me to learn more?
(Other definitions for nearby that I've seen before include "Just round the corner?" , "[OPPOSITE OF] miles away" , "Close at hand, not far away" , "somewhere close" , "Adjacent" .)