mail in icloud

Hey Babybear5000,

Welcome to Apple Support Communities. Time Machine should  back up your personal data, including apps, music, photos, email, and documents. For more Information, review this article: Back up your Mac with Time Machine. If they are iCloud emails, you may not see the option in Time Machine since they are synced with iCloud.

Is the 3rd email address an iCloud account or is it just a Gmail email? If it's an email, are you selecting Google as the email type?

Could you provide us the steps you take when adding the email? Here's a great article that shows steps on adding an email account: Add email accounts in Mail on Mac

Take care.

Hey Babybear5000,

At this point, we recommend contacting Apple Support for further help: Get Support

Take care.

Hey Babybear5000,

Are you trying to recover emails you’ve deleted in the Mail app on your Mac? If so, you’ll want to use the steps in this article to recover individual emails: Restore items backed up with Time Machine on Mac - Apple Support

Please let us know the results when you're able.

All the best.

Hi Babybear5000,

Thanks for following up. We would like have you reach out for further support here: Contact - How to Contact Us - Apple

Thanks.

Hi Babybear5000,

Don't hesitate to let us know if you we can further assist.

Cheers.