Hello 3Decademacuser,
Thank you for using Apple Support Communities.
We understand that you accidentally clicked on Apple Podcasts and closed it without further action. Now you are getting repeated requests asking for the password for Podcasts. It appears that your computer thinks you still want to log in to Podcasts, but for some reason the process has been interrupted. The easiest way is to complete the process and sign in to Podcasts. Once you have signed in, you can quit the app again and not bother with it any more. Take a look at this article to sign in to Podcasts:
Sign in to Podcasts on Mac
Best regards.