Error code -10 means the program online connection had an error, or was dropped/interupted.
Assuming the new computer is running Windows 11.
If the serial number begins with P2- then this may be the issue, the P2- serial numbers were deprecated over 10 years ago and used an older connection protocol and/or may no longer work.
You can try the following:
Turn on the indicated settings shown below, to get to them press the Windows key and the R key to open the Run dialog and type inetcpl.cpl and click OK.
Check that your Antivirus prgram is not blocking Music Maker accessing to the Internet, how you do this depends on the Av you are using.
If the above do not work then reverse the above settings, they are no longer needed for Windows 11 compatible software, and it may be time to consider upgrading to the current version, you should be able to get this for the Upgrade price.
Another potential issue with Windows 11 will arise if you logon to the computer with a Microsoft account. By default this installs and activates OneDrive, your Documents folder is moved to the OneDrive cloud storage.
This can affect programs such as Music Maker which read/write temporary files to the computer disc/ssd drive, with OneDrive active this can introduce delays while the files are written to and read from OneDrive. It also means you must always be connected to the Internet to access any files stored in your Documents folder when using Music Maker and other programs which create temporary files eg video editors.
As a minimum uploading to OneDrive must be disabled for the Magix folder in the Documents folder - see this article on how to do this.
If you do not need OneDrive, I would suggest you to disable it completely, the linked to article above has a link on how to do this.
If you want to uninstall OneDrive, you do so at your own risk and do ensure you have local backups of all documents/files etc before disconnecting/disabling OneDrive, you should still be able to access them through a web browser.