Being late last month was a relief (8)
I believe the answer is:
deceased
'being late' is the definition.
(I've seen this before)
'last month was a relief' is the wordplay.
'last month' becomes 'dec' (short for December, last month of the year).
'was a relief' becomes 'eased' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'dec'+'eased'='DECEASED'
Can you help me to learn more?
(Other definitions for deceased that I've seen before include "Defunct" , "departed" , "Recently dead" , "Pushing up the daisies" , "Passed away" .)