Retrieving deleted email

Log into the web interface from your email provider, and see if the messages are there. If they're not there, well, the mail messages may be gone, unless you have backups available via Time Machine or otherwise.

With OS X and its Mail.app, try rebuilding the mailbox. Use Mailbox > Rebuild from the menus.

If you get errors rebuilding the mailbox, use the Connection Doctor to verify the settings. Use Window > Connection Doctor to run a pass of the test, and if you get errors with that then open the drawer in the Connection Doctor window and re-run the tests to get some details of the failure. Expunge any personal information from those results, and post the output here.

I would also strongly recommend getting Time Machine going here, if you don't already have that configured and started. Having backups is the best way to recover from loss or from software or hardware failures. (If you do have Time Machine running and if the messages are gone from the server, you may be able to recover at least some of those via Time Machine.)

Yes, Time Machine is your first target. You need some patience in working within the small screen it allows you but if you know the operative dates, it's quite easy.