Deleted archive file

If the Archive e-mail were " On My Mac " refer to picture below - they " Archive ) are probably gone.

The Archived process removes the email from the IMAP configure Account and places them Locally as the " On My Mac" infers.

Any e-mails still on the IMAP account or nee e-mail can and should reappear.

Via web search:

Where are archived messages stored?

As we mentioned earlier, copies of your email messages are stored locally on your Mac when you use the Archive feature.

Unfortunately, they’re just not that easy to get to. They’re buried in a folder in the macOS Library and the archive itself uses a .mbox file extension.

You can access with the following steps.

• Open a Finder window.
• Hold down Option and click on Go in the top menu bar.
• Select Library from the dropdown menu.
• Look for a folder called Mail.
• From here, you should see another couple of folders that appear to be randomly named (V6 and a string of characters).
• Click through these folders until you find several .mbox. Click on Archive.
• From here, you can dive deeper into the folder tree until you see a folder named Messages. This is where macOS stores .emix copes of your emails in Archive.

There may be formatting or image issues, particularly if the content is web-based. But these are locally stored copies of your email messages. That, of course, means that they are immune to any server-based activities.

Being stored locally, cloud retrieval is not an option. But they may be in your TimeMachine backup.

Correct me if I understand correctly - ok!

The Older computer has the same setup " Archive " and "On My Mac ".

If that is true, the Archive and On My Mac E-mails on the older computer are safe when Wifi is connect on that computer.

In good conscience - suggest doing at Time Machine Backup on the Older Computer Before turning on the Wifi.

Welcome

You are welcome and thank you too.

Do come back to the Apple Support Communities ( ASC )  in the future, if / or when additional question may arise