Pegasus Mail stores numerous paths (home mailbox directory ,new mailbox directory, programs directory) in configuration files which are stored in a mailbox directory (eg: ...\MAIL\ADMIN). In order for a copy from one machine to another to be successful the paths must be identical.
I assume you can tell from the image what the path to the \PMAIL directory was when it was made. If it was anywhere inside of a \PROGRAM FILES directory don't read an further because this procedure won't work. Post back for an alternative approach.
If not inside of a \PROGRAM FILES directory here's what I suggest:
Rename the new C:\PMAIL take it of the picture for now (hopefully to be deleted).
Restore the backup to the same location.
Run C:\PMAIL\PROGRAMS\WINPM-32.EXE. This will start Pegasus Mail and create a registry entry of its path. You should be able to see folders and any new messages that existed when the backup was made. If you don't, something is wrong so stop here and post back, otherwise just close Pegasus Mail.
Run the installer. It will use the registry key to identify this Pegasus Mail instance and offer to upgrade it, which you should do. This will create shortcuts and other registry entries. You don't mention what version is being used but I suggest upgrading to the latest version (4.73).
Note: while doing this I think you might need to do it as the Win10 user who will be using Pegasus Mail, with that users privileges elevated to Admin. You should reduce them once the upgrade is complete.
Pegasus Mail should now function like it did when the image was made. The next step will be to address whatever isn't working. Let us know those details as you identify them and we'll help however we can.