Quote "I have purchased a copy of Lightroom"
Does this mean you have a 24 numerical digit License? If so then you need to uninstall the existing Lightroom Creative Cloud Application from your computer using the normal procedure for your operating system for a Mac you need to drag the Lightroom Application from your Applications folder into the trash.
You then proceed to install the latest update from the link below, proceed to install and activate by using your license key.
Quote "I have a free Adobe Cloud"
I do not think there is a Free Adobe Cloud what you probably have is a free 30 day trial of Lightroom installed which is why you are seeing the message to update your membership. The 30 day trial has expired.