Need email for my domain that works and syncs with mac mail

I am presently using Network Solutions as a provider for my webmail. It supposedly syncs with my Mac Mail, but does not, really. For instance, it usually rejects attempts to delete messages, popping up to say that the message could not be moved to trash.

I have three mail accounts in Mac Mail at this time: [name]@mac.com; [name.name]@[mydomain].com; and, a gmail account (necessary to sync with a cloud service I use for business calendaring). The only one I have trouble with is the middle one.

Any suggestions?

Thanks.