Hello @Howie411
What is the product number of your Omen so I can better know what you have? Open the HP Support Assistant to find the product number.
What size is the HDD?
The activation key is registered to the motherboard. You can reinstall Windows 10 as many times as you want on that machine, no matter what hard drive. No key is need during installation as Windows will self activate when it goes online to update.
Its best to burn a copy of your HP Recovery Media from the recovery partition for a safety backup.
You can access that process from Start > HP Help and Support > HP Recovery Media Creation. You would need a 16 GB USB pen drive to burn to.
If your SSD is the same capacity or larger than your original HDD, the HP media will install to it. Disconnect the original HDD first. Afterward, you will need to remove the OS on the HDD if it conflicts with the desktops ability to boot to the SSD.
Or
You can install Win10 from Microsoft's media creation tool. Install it to your original system, then use it to create a bootable USB drive (it can be between 4-32 GB).
Disconnect the original HDD and boot from the USB drive and install.
Then install whatever drivers are needed from the HP Support page for your model.
Let me know if you have further questions.