As I said above, with Win 10 a shutdown 'hibernates' any running programmes/processes and starts them again at the next boot.
You need to do a Restart to actually close any running programmes and processes (so that they won’t start again).
That is counter-intuitive, but it's the way Win 10 works.
So you are saying that Task Manager does not show anything FFx running, but CC says that it is? Strange.
I can only think that either something connected to FFx is running but has a name you can't recognise , or FFx has left some file open (or locked) so CC thinks that the whole of FFx is still open.
That can happen if Firefox closes unexpectedly.
(There is also a slight possibility that your Firefox user profile has become corrupted, but usually that won't let you open Firefox at all and Firefox itself tells you it's already running when you try to open it).
When Firefox is behaving strangely extensions or addons are usually the first thing to suspect.
First of all I'd try opening and closing Firefox and then Restarting (not shutting down) your computer, then without opening Firefox do an Analyze in CCleaner to see if it still thinks Firefox is open.
If CC says it's now closed then open FFX, visit a couple of websites, close FFX, and run a CC analyze again.
If CC now says it's still open then it shows that Firefox, or one of the addons, is not closing properly.
If something isn't closing properly I'd then try running Firefox in Safe Mode to rule out an extension being the problem.
https://support.mozilla.org/en-US/kb/troubleshoot-firefox-issues-using-safe-mode
Open Firefox in safe mode, visit a couple of websites, close it again, and do an analyze with CCleaner to see if it thinks it's still open.
If CC says it's closed do then it's almost certainly one of the addons causing the problem.
If that's the case then you should remove them and put them back one by one until you find which one is the problem.
If it's not the addons and it's still not fixed then I'd go for a 'Refresh Firefox'.
Note - A refresh will remove your addons (make a note of them first) and put all settings back to default, but it will keep your bookmarks and saved passwords.
https://support.mozilla.org/en-US/kb/refresh-firefox-reset-add-ons-and-settings
Of course there is always the option of uninstalling/reinstalling Firefox, but that's more drastic as it will lose all you bookmarks, and saved passwords.