I'm unable to update macOS 10.15.6

You might do the this: Download and Install macOS Catalina 10.15.6 Combo Update. You should run First Aid with Disk Utility before trying to update because you did an abnormal shutdown.

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.

You should always maintain at least 15-20GBs of free space. However, an installation requires much more during the installation. For example, Catalina is almost 5GBs. You need at least that much space for the Installer file. You then need space for the file to expand from the archive and space to keep your old system. Thus, allow about three times the size of the Installer. In this case, about 15GBs will be needed during installation. After the installation, the Installer will be automatically deleted, the new files overwrite the old files, and if all goes OK, the copy of the old system is deleted. If you need to free up some space:

How to Free Up Space on The Hard Drive

1. You can remove data from your Home folder except for the /Home/Library/ folder.
2. Visit The XLab FAQs and read the FAQ on freeing up space on your hard drive.
3. Also, see Freeing space on your Mac OS X startup disk.
4. What is 'Other' and 'Purgeable' in About This Mac?
5. Files That Make Up the 'Other' Storage Category, and How to Remove Them
6. Free up storage space on your Mac.
7. See Where did my Disk Space go?.
8. Be sure to Empty the Trash to recover the space.
9. Replace the drive with a larger one. Check out OWC for drives, tutorials, and toolkits.
10. Use OmniDiskSweeper or GrandPerspective to search your drive for large files and where they are located.

Get Correct Storage Information

Do not use the information from the Storage section of the About This Mac dialog. Ignore the Storage information as it is typically wrong. To find out the correct information for any disk: Select a Desktop disk icon. Press Command-I to open the Get Info window and look at the topmost panel displayed. You will find the disk information displayed for Capacity, Available, and Used. If you have more than one disk/partition then repeat for each one on your Desktop.

The categories found in the Storage section of About This Mac is simply an arbitrary way of displaying files on your drive. There are no such categories actually on the drive.

Wow, this is just broken. I have an older MacBook Pro that I always update first and this is why. I always download the combo update from Apple Downloads and I did that a little while ago. I then started the update on my older 2013 MacBook Pro. First thing I notice is the amount of time before it says "ready to install, please restart" or something like that, is quite a bit longer than what I am used to in my zillion previous updates.

Then, just as others are reporting, it restarts, moves a progress bar the tiniest amount in a very short time, then I see the screen go black again, another restart, only this time it simply is a restart to boot back into my previous macOS which is 10.15.5.

This is just broken and many others are reporting the same. I am not skilled enough nor do I have the desire to try to fix Apple's broken software... Too many people are reporting the same so I hope Apple gets the message that they put out an installer that , for at least many, does NOT work.

Still reboots fine into 10.15.5 but just seems to try to start the 10.15.6 update. That's the tiny progress bar and for at most 2-3 seconds, and then it just quits the update and reboots the computer into the old 10.15.5... BROKEN!!!! I will wait...

Yes, felicia279 (and Robert Paris, and, perhaps, mon1225):

This is what we've been seeing with this update: insufficient disk space.

Now. It is certainly not your fault that this update is not informing users of the lack of disk space, or of how much disk space is needed!

Fortunately, those of us that have seen this enough know that you need something more than twenty some odd Giga Bytes of free storage for this update. (The actual amount probably varies with hardware, and, maybe, other aspects of the current system.)

Kappy and Halliday,, YES you were right. I was stunned to see how much data I still had on an older (2013) MacBook Pro and yes, that is why it would NOT update. I found the 500 GByte hard drive to have ONLY 17 GBytes free. So 17 was not enough. I pretty easily cleaned off an old VMware Windows file that brought my free space up to 88 GBytes. And where twice yesterday, the update would just quit and go back to my normal logon, today, with 88 free GBytes, it is updating right now and is just about to get done. Thanks for the help... Oh, and just to be complete, I always do updates by downloading the entire Combo update. I did that here as well... Combo Update... thanks again...

I'm having this same problem right now too. The internet browser opens up at the start up; after trying to update.