How do I get Mozilla to stop asking me for access to my google account?

If you want Thunderbird to manage your Gmail mail, you have to tell Gmail that its OK with you. By continually choosing "deny", you're telling Gmail that it's NOT OK for Thunderbird to access your email on their server. Therefore, Thunderbird will never be able to receive or send any.

The way to make it go away is

a) agree to let Thunderbird manage your email for you, so you can use Thunderbird's extensive email management tools and features, or...

b) to not use any local email agent, be it Thunderbird, Outlook, or whatever. Your only choice then would be to stick to web mail.

Thunderbird already manages one of my accounts. Do I need it to manage my backup also (I use it only when the primary one is unavailable).

Thanks for your reply. Larry

I don't think I understand your question, but in general, every account in Thunderbird that fetches or sends email must be authorized by the email provider in order to do its work.

The only "exception" is the Local Folders account, which is not associated with an email provider and never reaches out to the Internet.