My mac is set to a different language, so apologies if some of the terms are not totally correct, but go to your system settings, choose safety and anonymity. At the bottom you should see an option to select which programme should be opened. If you tried to open PlayOnMac, it should give you an option to open regardless of developer.
First click ok. Then you need to open System Preferences. On the general tab you need unlock to make changes, then mark that it is ok to run a program from the developer. After you do that, if you have a previous OS to Catalina, it will open fine. If you have Catalina, you have to wait for the PlayOnMac developers to make a version compatible with Catalina. 32 bit programs will not run on Catalina, and neither will Wine :( I just got a new computer, and I cannot install an earlier operating system. If I want to use PlayOnMac, I have to use my old computer to do so.