I've always had my browsers clean themselves on closing and then also used CCleaner. If Firefox worked that way; that is what I'm accustomed to and I wouldn't have bothered asking for a solution from CCleaner. Firefox doesn't anymore and apparently has not for a long time. Your history remains in "places.sqlite".
You cannot clean "places.sqlite" on exit and if you delete it it will just be regenerated with all the history you want to get rid off. It also contains your bookmarks that you don't want to get rid of.
The attachment is from Mozilla and says " Places.sqlite is used in Firefox 3 and above, instead of the older bookmarks.html and history.dat files" and that the file "places.sqlite" stores the annotations, bookmarks, favorite icons, input history, keywords, and browsing history (a record of visited pages). Seeing that we are on Firefox 63 I'd say CCleaner has had plenty of time to identify and deal with this issue.
There is a difference between telling Firefox not to display info in the Awesome Bar aka the address bar, which you can do, and getting it off your disk which means getting it out of "places.sqlite" which you can't do.
The contents of "places.sqlite" can be examined in several ways described in the attached article yielding the annotations, bookmarks, favorite icons, input history, keywords, and browsing history (a record of visited pages).
CCleaner says it cleans the internet history and the download history among other things (see attached image) which it cannot do unless it cleans "places.sqlite" which it can't do because Firefox won't let it.
"places.sqlite" is a 5MB file created during installation by Firefox. "places.sqlite" regenerates itself as a 5MB file containing all your histories by default if you delete it.
I'd just like to use CCleaner with the confidence that it is doing what it says. In this instance it is not. I'd really like CCleaner to solve this.
Places.sqlite - in Firefox stores the annotations, bookmarks, favorite icons, input history, keywords, and browsing history (a record of visited pages).html