Mail not syncing properly inside folders

Can you give information about which kind of email account you use? For example, are you using an iCloud, Gmail, Exchange account or another party? Do you know whether you set it up with IMAP or POP? If your email account uses POP, this article can be usefull: Get help if you have issues using a POP email account with more than one device - Apple Support

Additional information about your current setup would be useful.

If the problems arose after you used the 'cleaner' then this 'cleaner' probably is the cause and has something removed that shouldn't have done it. Since we don't know what the 'cleaner' has removed, it will be difficult to figure out where the problem lies.

If you know the name of the cleaner is, then probably some user has suggestions or experience with it, not sure.

This article is not useful for solving the problem but it explains, among other things, why you shouldn't use 'cleaners': "Never install any product that claims to "clean up", "speed up", "optimize", "boost" or "accelerate" your Mac; to "wash" it, "tune" it, or to make it "shiny"." Effective defenses against malware and other threats

Another approach to try what hopefully helps, assuming what you've said, settings should be ok.

Start your system in safe mode, see Paragraph "Start up in safe mode":

Use safe mode to isolate issues with your Mac - Apple Support

We only use safe mode for clearing caches so after login, wait a minute or so then reboot normally.

After normal startup, start Mail and don't select any mailbox while mail gets (or not) new messages. Check Mail activity, choose Window > Activity, then watch the progress if there is any. When there is no more activity, we goto next step.

Leave Activity Window open, it will stay on top, you can drag it around if needed. Select your mailbox in the Mail sidebar, then choose Mailbox > Rebuild.

When rebuilding you Exchange account, messages and attachments stored on your computer are discarded, and then downloaded again from the mail server to your Mac. Your mailbox appears empty until the download is complete. Only look at the Activity Window and leave 'mail alone' until there is no more activity. If this works out, you should have working mail again.

As long as your MS Exchange account does not have a good, complete connection to the server, mail will not properly synchronise including moving and deleting mails.

Are you able to contact the system administrator of the MS Exchange server to ask for the right settings? It also depends on how the MS Exchange server is configured.

Malwarebytes wouldn't have had anything to do with this. It doesn't touch Mail-related stuff. I wouldn't call it a "cleaner," as that word has different connotations and associations than what Malwarebytes does.

Exchange support in Mail isn't perfect, and can be kind of flaky sometimes, in my experience. You'll definitely need to talk to your IT folks, as this sounds like a configuration issue on your Mac.

To setup an MS Exchange account be sure to follow the procedure as described in this article: Use Microsoft Exchange (EWS) accounts in Mail on your Mac - Apple Support

Agree, even though I do not use it myself. Hope that another user who has experience with MalwareBytes can say something about what (maybe) it can do on the MS Exchange account/settings.