I have only come across this once or twice. In that case, the site was not recognizing the Screen Reader, but a javascript plugin for the site kept updating a lie region to say something like "for better screen reader response use our plugin. So I turned on the plugin, and the noises went away, but the site was pretty horrible to use after that. 
JAWS and VoiceOVer have ways to ignore live region updates on noisy pages, so I expect that NVDA does too.
One other possibility... Could one block a specific JS package on a site? Or of course, you could tell your browser to not allow JS and that would prevent this extra mess.