I had the same problem. Once I checked the DNS query block log I found three queries that could be related to CCleaner. It seems the servers CCleaner uses got black listed by various DNS ad blocking lists which gets included on one's pi-hole.
I don't see why CCleaner is requiring to phone home when all it needs to do is clean up temporary files and other junk from my system. Is there a repository somewhere I can download previous versions of CCleaner that didn't have this issue?
EDIT: Aside from the license.piriform.com query, whitelisting the general amazon/avast servers is a no-go because it'll allow ads to come through on other services.