From my experience, this occurs a day or two after a password change. The steps below will resolve this problem with Outlook 2019, but I'm not sure about 2013. Below is a step-by-step guide I have used with my computer and computers of my co-workers. If you have any questions, let me know.
Email Issue: After you’ve signed in and checked the “remember me” box, the email still asks for your password. This not an uncommon problem and the fix is fairly simple.
Spoiler alert: this requires a REBOOT, so you’ll probably want to close everything else out.
Go to the control panel (click the windows button and start typing “control” (even if there is not a text box there) and it will pop up in the menu).
Go to “Credentials Manager”
Click on the “Windows Credentials” tab
Under the Generic Credentials section, remove all of the credentials. Don’t worry about removing them; Windows will repopulate them on reboot.
Reboot the computer. (Note: If the computer wants to reboot in Safe Mode, go ahead and let it, then reboot again after the final step and it should reboot normally.)
After reboot, open Outlook and click on the “File” tab
Click on the “Account Settings” box (first one), then click “Account Settings” on the dropdown menu
Make sure your email is highlighted and click the “Change” button.
On the next screen, click the “Next” button.
Try signing in again. It should work this time and, if you clicked the “remember” box, it should not prompt you again.
Thanks so much Novicegeek, but I can't do that as I think I've gone to far. Before your email, I tried deleting the email address completely with a view to starting it up again. Apparently that hasn't happened as the password prompt box keeps appearing although the email address has gone.
What to do now? Oh woe.
Sorry for the misinformation. That solution works beautifully for that issue with straight up Outlook 2019; however, in re-reading your email, it has finally dawned on me that you are accessing your email account with Outlook.
The good news is, all hope is not lost. I would make sure that POP/IMAP is enabled with both the Outlook and Gmail accounts.