to fix the problem, go to ccleaner's "options"/"advanced" and, at the bottom, click "restore default settings." after that, when you run ccleaner, if edge is running, you will see a prompt, asking if you want ccleaner to close edge.
as nergal said, another option is to set edge to where it does not have processes running in the background. the same applies to chrome and firefox: set them to where they don't have processes running in the background.
with edge, to stop its processes from running in the background, in edge's settings, go to "system and performance" and, there, turn off "startup boost" and "continue running background extensions and apps when microsoft edge is closed."
you also need to remove the "bing desktop search bar" from your computer-screen, if you have that on your computer, in order for edge-processes not to be running in the background. if the "bing desktop search bar" is displayed on your computer-screen, edge-processes will be running in the background, at least that was the case with my windows 10 computer.
here is an article about the "bing desktop search bar":
seems to be the same reg-tweak, in both cases, for disabling the desktop search bar.
note that, at least on windows 10, using the reg-tweak to disable the desktop search bar is not required. the desktop search bar itself has an option to simply "close" it, and it stays closed, where edge's background processes do not run in the background.
i wasn't very clear in what i was saying. normally, when you run ccleaner, if "edge" is running, you will see a prompt, asking you if you want ccleaner to close it.
you must have ticked the box for "do not show me this message again" which was why you were not seeing the prompt, asking you if you wanted ccleaner to close edge.
i think that clicking "restore default settings," in ccleaner's "options"/"advanced," would restore the default settings, where you again would see the prompt, asking you if you wanted ccleaner to close "edge."
you would see a similar prompt if "chrome" or "firefox" was running.