I purchased a yearly Lightroom subscription using private relay. How do I restore the purchase on my mac? I have the private relay email.

Do you mean Apple iCloud Private Relay? If so, that doesn’t affect how you get into your subscription because Private Relay is more like a virtual private network (VPN); it doesn’t really change how email works.

Maybe, do you mean you used Apple Hide My Email to purchase the subscription? That is much more likely. If that means your Lightroom subscription is associated with an Apple Hide My Email address, then all you have to do is use that email address to sign into the Adobe Creative Cloud desktop app, and you will have access to the software that your subscription covers.