Hello Originaladrian and welcome to Apple Support Communities.
I see you are having an issue with email not syncing between a Mac and an iPhone. From the way this is phrased it sounds like this is less about the "read/unread" status and more about filing individual messages into mailboxes.
Get help using iCloud Mail - Apple Support
If this is regarding an iCloud mail account, you'll want to start by signing out of both devices. (On the Mac, do that at System Preferences > iCloud. On the iPhone do it at Settings > [your name] > iCloud. Next shut down both devices. Then reboot the Mac and sign in and finally, do the same on the iPhone.
If that process does not resolve the issue or if this is not an iCloud account, then delete the account, reboot and then add the account back. As long as this is an account hosted by a major service such as Google or Yahoo, the account should be of the IMAP type so you do not risk data loss as messages live on their server, not just on your device. If you are unsure, then check with your provider before taking this step.
Add or remove email accounts in Mail on Mac - Apple Support
Add an email account to your iPhone, iPad, or iPod touch - Apple Support
Cheers.