How do I get the HD-DATA Disk to show up on my Mac mini desktop?

Okay. I believe I understand.

FYI - Disk Utility will show both – "HD#1" and "HD#1 - Data" – as volumes on that startup drive.

Finder never will show "HD#1 - Data" on the desktop. The macOS manages both of those volumes together and will only ever show you "HD#1" on the desktop as your startup drive. When you access that desktop icon, macOS is doing magic in the background and you actually are storing your user files on the "... - Data" volume. The other volume is a secured, read-only volume that houses the OS files, keeping them safe (hopefully) from tampering and corruption.

Why you have no write access now to your startup drive is very concerning.

Please reboot the Mac in Recovery mode (⌘R) and use Disk Utility's First Aid on those volumes and the startup device as well. You must be booted in Recovery to run the full routines on that drive, some corrections cannot be made when booted from your regular startup drive.

Please see: How to repair a Mac disk with Disk Utility - Apple Support

When you launch Disk Utility, make sure that you select to View All Devices so that you can select the volumes and then also the device those volumes are installed on.

As I said, to see the HD on the desktop you only need to go to Finder > Preferences > General tab and check mark the box to Show Hard Disks on the desktop. Have you done this?

All of your files are stored on the HD that Finder shows you at the top right corner of the desktop.

I feel that there's something here that I'm missing. It will be helpful if you would describe exactly what you are trying to accomplish.

The HD that you are trying to see on your desktop, is it an external drive that you're trying to work with?

You don't. It is part of a set of two APFS volumes that the Finder and macOS treat as a single volume called (usually) Macintosh HD and it will never be displayed as a separate drive volume.

If you have Macintosh HD on your desktop, then you are basically looking at the Macintosh HD - Data volume that houses your Home folder and all of your user account data and settings.

To view Macintosh HD on the desktop, go to Finder > Preferences > General and mark the check box to Show Hard Disks on the desktop.