OK, thanks for the heads-up. Separate user accounts is still probably the right way forward, though.
FYI, v5.56.7144 did not clear Firefox history or cache this morning, so the problem (for me, anyway) persists.
It does work fine when I run it manually. So it's as if Firefox is unavailable to CC when the machine boots, rather than CC is unable to clean the right areas.
Anyway, I'll head off to configure the browser setting/