macOS Catalina 10.15.6 Update

Try updating/downloading through the Apple Store and NOT from the System Preferences>Software Update. This worked for me just many hours ago. I asked here about downloading/installing macOS Catalina 10.15.6 and Kappy had this reply below which was very helpful. I am now running on macOS Catalina 10.15.6. Hope this will work for you.

Repair the Drive for El Capitan or Later

1. Restart the computer. Immediately, at or before the chime, hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
2. Select Disk Utility and press the Continue button.
3. Then select the indented (usually, Macintosh HD) volume entry from the side list. 
4. Click on the First Aid button in the toolbar and wait for the Done button to activate. Click on it, then quit Disk Utility.
5. Select Restart from the Apple menu.

Also, see the following:

Before You Upgrade macOS

1. Fix any problems before upgrading.
2. Run First Aid in Disk Utility while booted from your Recovery HD.
3. Backup. Make a bootable clone using CarbonCopy Cloner or use TimeMachine.
4. Be sure your startup disk has 18-20GB of free space during the installation.
5. Upgrade third-party software for compatibility. See App Compatibility Table - RoaringApps. If you have mission critical software, then DO NOT upgrade until you know that software is compatible with the version of macOS you are about to install.
6. Disconnect all third-party peripherals.
7. Download the installer which will be in the Applications folder. Quit the installer after it opens, then make a copy of the installer and save it in the Downloads folder. The one in the Applications folder will be deleted after a successful installation.
8. The download is large, anywhere between 5 and 9GBs. It takes time to complete. Be patient.
9. Double-click on the installer in the Applications folder. Click on the start button to begin the installation.

Instead of using System Preferences> Software Update, and by the way never have it set to update automatically.

Go to the Apple Support Downloads site,

Apple - Support - Downloads

Here you can see there are three available downloads for Catalina.

The 10.15.6 update, updates Catalina from 10.15.5 to 10.15.6.

The 10.15.6 Combo update, updates any previous version of Catalina to 10.15.6 (preferred)

And the 10.15.6 Supplemental update, do this one last.

Select which 10.15.6 update you want to download, the Combo is preferred as it resets some

system files and caches and once downloaded,. double-click to install.

Then you can download and install the Supplemental.

Whenever downloading any updates from Apple have your mac connected to your router via cable,

do not use your mac while it is downloading or any other devices that share the same internet connection.

Make sure you always have enough free space on your disk, keep at least 15-20GBs of free space

at all times. If you run your mac too low it can seize up and become unusable.

Which update did you try to install, did you download the Combo.

Remember you cannot install the Supplemental until you have already updated to 10.15.6.