Hi, is this problem on an iPhone/iPad (iOS) or on a Mac (MacOS)?

For Mac:

Generally speaking, the popup blocker works by preventing sites from launching completely separate new windows. When a site adds overlays within the page, that does not use the popup window mechanism.

When you click off the message, Google normally sets a cookie so that your choice is remembered. Of course, every time that cookie is cleared, you are a new visitor to them. Is it worth keeping Google cookies just to resolve this issue? Probably not.

So that leaves a possible add-on solution. You may find that a general purpose blocker like uBlock Origin can hide those suggestions, or maybe another one. I haven't tested myself.

https://addons.mozilla.org/firefox/addon/ublock-origin/

Thanks. I hadn’t considered that they wanted to leave a cookie to remember my answer. I downloaded that utility but can’t determine how to install it on my iPad or integrate it with the Fire Fox running there. Thanks again for the response. I’ll mark this closed because I was more looking for ideas than complaining about Fire Fox. I’ll direct the actual complaint at Google. Thanks again.

You shouldn't need to download anything to save cookies in Firefox.

Please be super-cautious about any apps that suggest they can modify Firefox since you often have personal data in there.