I think you want:
SELECT * FROM calendar
WHERE
(calendar.EventDate = DATE_ADD(CURDATE(),INTERVAL 1 MONTH) AND calendar.DateAdded < DATE_ADD(calendar.EventDate,INTERVAL 1 MONTH))
OR
(calendar.DateAdded = CURDATE() AND calendar.EventDate BETWEEN CURDATE() AND DATE_ADD(calendar.EventDate, INTERVAL 1 MONTH))