Annual that's soon after end of May (6)
I believe the answer is:
yearly
'annual' is the definition.
('yearly' can be a synonym of 'annual')
'soon after end of may' is the wordplay.
'soon' becomes 'early' (I've seen this before).
'after' is a charade indicator (letters next to each other).
'end of' suggests the final letters.
The final letter of 'may' is 'y'.
'early' put after 'y' is 'YEARLY'.
'that's' is the link.
(Other definitions for yearly that I've seen before include "Approximately every 52 weeks" , "At annual intervals" , "Perennial" , "on a regular basis" , "Every 365 or 366 days" .)