Two hard drive icons on desktop

A Troubleshooting Procedure that may Fix Problems with macOS El Capitan or Later

You should try each, one at a time, then test to see if the problem is fixed before going on to the next.

    Be sure to backup your files before proceeding if possible.

1. Shutdown the computer, wait 30 seconds, restart the computer.
2. Disconnect all third-party peripherals.
3. Resetting your Mac's PRAM and NVRAM
4. Reset the System Management Controller (SMC)
5. Reset your Startup Disk and Sound preferences.
6. Start the computer in Safe Mode. Test in safe mode to see if the problem persists, then restart normally.
7. Use Apple Hardware Test to see if there is any hardware malfunction.
8. Repair the disk by booting from the Recovery HD. Immediately after the chime hold down the Command and R keys until the Utility Menu appears. Choose Disk Utility and click on the Continue button. Select the indented (usually, Macintosh HD) volume entry from the side list.  Click on the First Aid button in the toolbar. Wait for the Done button to appear. Quit Disk Utility and return to the Utility Menu. Restart the computer from the Apple Menu.
9. Repair permissions on the Home folder: Resolve issues caused by changing the permissions of items in your home folder.
10. Create a New User Account Open Users & Groups preferences. Click on the lock icon and enter your Admin password when prompted. On the left under Current User click on the Add [+] button under Login Options. Setup a new Admin user account. Upon completion log out of your current account then log into the new account. If your problems cease, then consider switching to the new account and transferring your files to it - Transferring files from one User Account to another.
11. Download and install the OS X El Capitan 10.11.6 Combo Update or 10.12.6 Combo Update or Download macOS High Sierra 10.13.5 Combo Update or Get MacOS Mojave now from the Mac App Store as needed.
12. Reinstall OS X by booting from the Recovery HD using the Command and R keys. When the Utility Menu appears select Reinstall OS X then click on the Continue button.
13. Erase and Install OS X Restart the computer. Immediately after the chime hold down the Command and R keys until the Apple logo appears. When the Utility Menu appears:

     1. Select Disk Utility from the Utility Menu and click on the Continue button.

     2. When Disk Utility loads select the drive (out-dented entry) from the Device list.

     3. Click on the Erase icon in Disk Utility's toolbar. A panel will drop down.

     4. Set the Format type to APFS (for SSDs only) or Mac OS Extended (Journaled.)

     5. Click on the Apply button, then wait for the Done button to activate and click on it.

     6. Quit Disk Utility and return to the Utility Menu.

     7. Select Reinstall OS X and click on the Continue button.

Here are a couple of ideas. Right click or control click on the dubious hard drive icon and click on "Get Info". What does it tell you, how much data is on it. Where is it located. Open disk Utility and run First Aid on it and see what happens. Try a restart or even a Safe Mode restart and see if this might help as was already mentioned.

" I was hoping for some suggestions a little more specific"

Finder>Preferences>General>Hard Disk —uncheck the box

Have you tried changing its permissions so you can view its contents and then hopefully determine exactly what is going on?

Mine does. This only started after installing the Mojave 10.14.3 Supplemental Update.

Before you dismiss a suggestion you know nothing about perhaps you first should try it. To be perfectly honest, anyone who begins a sentence with, "Being honest," isn't really being honest. They are being dismissive.

Kappy, before you make assumptions about what someone knows or whether they've tried your suggestion, perhaps you should first ask them before you accuse them of being dismissive. What's dismissive is posting a generic, one-size-fits-all troubleshooting script that not only doesn't address the question that was asked, but ultimately suggests an advanced procedure (wiping the hard drive and reinstalling the operating system) to correct an annoying glitch!

Ha, thanks, but that wont get me to where I wanna be.

And where exactly do you want to be?

If you —uncheck the box in preferences and test in this environment— does your mysterious 2nd HD icon reappear?

Seems it has something to do with a Time Machine backup session

Continued from above:

14. If none of the above helps then see How to Downgrade macOS High Sierra and macOS Reversion- 

      How to Downgrade from High Sierra.

15. If you get here without success then it's time to make an appointment at the Apple Genius Bar for 

      service. If you need to find an Apple Store - Find a Store - Apple.

I always see this. Why? Because I have two partitions on my disk: One is for storage, and that does take a moment to show up on my screen when I boot.

I don’t think this would be a partition, but to find out, use Disk Utility to reveal how many partitions you may have:

Go To: Applications > Utilities > Disk Utility

If only one shows, click on the info button (located on the top-right corner of the Disk Utility window). What is the Partition number quantity? And how many disks are shown?