I'm using CPanel with Exim. Is there a way to manually add email addresses to Exim?

Yes, this is possible. You can use /scripts/addpop to add email accounts from the command line. The addpop script will take the following variables:

/scripts/addpop email@domain.com password quota

For example, say you have a domain example.com and want to create an email account "admin" with a password "h3LL0th3r3" and a quota of 1000 MB, you would run the following:

/scripts/addpop admin@example.com h3LL0th3r3 1000

This script will create a working email account that will show up in cPanel > Email Accounts area.