I forgot to say that I am not the actual administrator of the system.
I'm a customer of a hosting provider, namely, ovica.com as you can see
in the headers.
I've emailed them about it several days ago, but was hoping to speed
things up by also asking on the Exim list, in case it is a well-known
issue.
So... unfortunately, I only have cPanel access. :p
The only thing that I can tell is that Exim receives the incoming
message on time, but only tacks on the "Delivery-date:" header after
many hours, and it happens to an apparently-random minority of emails.
Any thoughts?
If I don't hear from my provider soon, I'll forward your suggestions
about checking the logs to them.