MacOS Catalina update failing due to low disk space

Go in the Recovery menu (hold cmd+r), in the top left corner, you should see the Apple logo. Click on it and select the boot disk, which should appear as your internal hard disk, you will be prompted to enter your password to decrypt it and you should be good to boot into your old system.

There certainly is.
I believe that I have worked out one of the best ways of going about such a task as I found myself having to do this frequently due to problems similar to those you are experiencing. I may just add that they stopped after the macOS High Sierra upgrade in September of 2017.

Go directly into the Mac AppStore's temporary files folder:

1. Quit the Mac AppStore completely (no updates running etc.)
2. Open Terminal and type: open $TMPDIR../C/com.apple.appstore/, hit the Return key.
3. A Finder window should have popped up with the files and folders in the "com.apple.appstore" location. Move (not copy) these to another folder (perhaps make one on the desktop) in case things go sideways, or (if you are confident), move them to the Trash right away.
4. Most importantly, do not delete or adjust any other files outside of this directory, when finished close the "com.apple.applestore" folder.
5. Relaunch the App Store.

Verification and several other download issues should be fixed with these steps and you may find that you are once again able to properly download and install the applications fully (i.e. not starting midway).

What next?
I would further suggest running one or two sweeps of CCleaner for Mac (a free, trusted software) with options to fine-tune specific types of files.

1. Open CCleaner

2. Go to Cleaner, select the Applications tab

3. Under Applications, check App Store

   If CCleaner is run after completing clearing the AppStore temp files as described in the first set of steps, CCleaner will most likely remove any residual files and folders from the App Store that are only half downloaded or otherwise exhibit similar properties. It just works!

For an additional read, this article explains my thinking behind the first suggestion, while this one may be a little more general.
Let me know how you make out or if there is anything you would like me to clarify. I'm always more than happy to do into further details.