Hi CCLou83,
Thanks for following up. This link has some great general information About missing information after you restore your iPhone, iPad, or iPod touch with iCloud Backup.
About your Mail and Notes data
Since your Mail already lives in iCloud, it isn't included in iCloud Backup. iCloud Backup does include the settings for your iCloud email account and any other email accounts on your device. After you restore from iCloud Backup, iCloud and your other email providers replace your available email messages so you can pick up where you left off.
If you set up your device to store Notes in iCloud, iCloud sends those notes to your device after the restore process finishes. To find storage options for Notes in iOS 10.3 or later, tap Settings > [your name] > iCloud.
Notes data stored by a third-party service (like Microsoft Exchange) isn't backed up to iCloud. Contact the provider of that service for information about how to retrieve your data.
You mentioned you see the emails in question on another device. Are you using a POP email account, or IMAP? If you're using POP, this might be normal, as emails are downloaded from the server rather than syncing across all devices.
Let us know!