Dear xxx,
I am really sorry for this belated e-mail.
To tell the truth, I have finished the work for the contest quite some time before, but due to the busy graduation season, I put it aside and forgot all about it until now. I thought there might be an extension of the deadline like that of last year. I also considered not sending the work at all as I've already missed the deadline. Eventually, I made up my mind in the belief that sending it late is better than not sending it at all.
I sincerely hope that you could accept my work as if it didn't miss the deadline.
I am looking forward to your kind considerations.
Many thanks in advance.
Yours Sincerely,
xxx