How do I cancel notifications that say, "1 Problem Found on Network"

Maybe the settings suggested by DavidR may reduce the unwanted announcements. See here: How to block the AVAST adverts - #2 by DavidR

Also check if Smart Scan is being performed as a scheduled scan in Avast “Virus Scan” settings. That would display repeated announcements, but not every 20mins. It is set scheduled ON as default, but you can turn that OFF.