Thanks for reporting this.
Is this a new behaviour that only started this week?
Have you updated to Firefox 85.0?
There have been a lot of changes 'under the hood' in Firefox 85.0 which was released this week - some of which change the way tabs, networking, and caches work in the background.
https://blog.mozilla.org/security/2021/01/26/supercookie-protections/
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In Firefox 85, we’re introducing a fundamental change in the browser’s network architecture to make all of our users safer: we now partition network connections and caches by the website being visited.
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They have also added some 'fingerprinting' protection which may be more relevant to the issue of your accessing websites.
https://blog.mozilla.org/security/2020/01/07/firefox-72-fingerprinting/
When Mozilla make changes like this it can sometimes take others, such as CCleaner, time to catch up with those changes.
(We have seen this in the past, most recently when they split out the Startup Cache a few months ago).
I've not seen any odd effects in Firefox 85.0 yet myself when cleaning Firefox 85.0 with CCleaner, but different users use their browsers in different ways.
It is something I'd been watching for just in case, and yours is the first report here of any issues since Firefox 85.0 was released.
Just for more info could you tell us:
Your Windows version. (eg. Windows 10 Home 20H2)
Your CCleaner version.
Which Antivirus you are using.