Welcome to iMore! Please test after each step.
1.Go to Settings > Mail, Contacts, Calendars > Your Account > Mail > Turn Mail off and then, wait a few seconds, back on.
2.You may try deleting the email account from your iPhone or iPad. Just tap Settings > Mail, Contacts, Calendars > Your Account > and tap Delete Account. Then re-add your account by going to Settings > Mail, Contacts, Calendars > Add Account.
3.Tap Settings > Mail, Contacts, Calendar > Your Account > Account > Advanced > Deleted Mailbox and set this to trash then tap Advanced > Account > Done.
4.Tap Settings > Mail, Contacts, Calendar > Your Account > Account > Advanced > IMAP Path Prefix, then enter Inbox; when you done, tap Account > Done.
6.Open your Mail app on your iOS device, then tap Index > Edit > find the email messages that you can?t move/delete and then select them > tap Mark and Mark as Read (or Unread).