I 'may' have found a solution, at least to the 5.38.6357 (64-bit) version on Win 7.
I opened Task Scheduler and disabled then deleted the CCleaner update entry, which shouldn't have been there as I have it disabled in CCleaner itself...
The I opened regedit and searched for Piriform (match whole string only), and for every result, I checked the nested CCleaner key and it's entries.
Any that said update I deleted, and any that said 'monitor' or 'monitoring', I changed the value from 1 to zero (0). All monitoring was also disabled in CCleaner, before I did this.
Any way, I then rebooted and so far, I haven't seen any new CCleaner entries in Task Manager.
Those entries that appeared when the program was NOT running have seemed to stay away. Usually they would return after like... 12 hours, but so far they haven't. I have run the program 3 or 4 times since I made those changes in Task Scheduler and the Registry. If the problem comes back I will update this topic.
/edit Well it seems that the adjustments I made, have solved the problem. It's now the 4th day since the adjustments and still no issues.