Edit: To summarize, I purged the packages and installed python3.10 as a new package. According to others who followed this, you do not need to remove and add the ppa as I did. I am removing that from the answer. It has also been suggested that merely forcing the install with sudo apt install -yf has worked when this issue came up in past versions. I can't verify because I did the following instead:

I had installed Python 3.10 using deadsnakes.

The issue and solution are described here:

libpython3.10-minimal and libpython3.10-stdlib fail to install #207

I ran the suggested command (explained in more detail at the end of this answer):

sudo apt --fix-missing purge $(dpkg -l | grep 'python3.1[01]' | awk '{print $2}')

This prompted with:

The following packages will be REMOVED: idle-python3.10*
libpython3.10-minimal* libpython3.10-stdlib*

libpython3.10-testsuite* python3.10* python3.10-distutils*

python3.10-examples* python3.10-full* python3.10-gdbm*
python3.10-lib2to3* python3.10-minimal* python3.10-tk*
python3.10-venv*

Note that there may be other packages removed with this command.

As suggested, I ran:

sudo apt --fix-broken install

There wasn't anything to fix because it had been purged.

Next I went ahead and upgraded unrelated packages before continuing:

sudo apt upgrade

At this point, running python --version showed it had rolled back to Python 3.8.10. I want Python 3.10 again so I ran:

sudo apt install python3.10

This prompted with:

The following additional packages will be installed:

libpython3.10-minimal libpython3.10-stdlib python3.10-minimal
Suggested packages: python3.10-venv The following NEW packages will
be installed: libpython3.10-minimal libpython3.10-stdlib python3.10
python3.10-minimal

The new install was a success!

Explanation For Cautious Beginners

The command used for purging python3.10 packages was:

sudo apt --fix-missing purge $(dpkg -l | grep 'python3.1[01]' | awk '{print $2}')

The description of --fix-missing is found in man -apt-get:

Ignore missing packages; if packages cannot be retrieved or fail
the integrity check after retrieval (corrupted package files), hold
back those packages and handle the result. Use of this option
together with -f may produce an error in some situations. If a
package is selected for installation (particularly if it is
mentioned on the command line) and it could not be downloaded then
it will be silently held back.

purge:

purge is identical to remove except that packages are removed and
purged (any configuration files are deleted too).

For the package names, a command substitution is used.

dpkg -l package-name-pattern...:

List packages matching given pattern.

Because no pattern was given for dpkg, a list of all installed packages is returned. In lieu of a pattern, the list is piped into grep so we can used the pattern 'python3.1[01]' to narrow the list down to installed python3.10 packages. These results are then piped into awk '{print $2}'.

awk '{print $2}':

To put it simply, this awk a pattern scanning command. Here it returns only the package name from each line in the list. To better understand, run these commands together and observe the output:

dpkg -l | grep 'python3.1[01]' | awk '{print $2}'

This should output a list of all of installed python3.10 package names, such as:

libpython3.10-minimal:amd64

libpython3.10-stdlib:amd64

python3.10

python3.10-distutils

python3.10-lib2to3

python3.10-minimal

python3.10-venv

The end result is the same as if you had entered all python3.10 packages yourself:

sudo apt --fix-missing purge libpython3.10-minimal:amd64 libpython3.10-stdlib:amd64 python3.10 python3.10-distutils python3.10-lib2to3 python3.10-minimal python3.10-venv

Now the system should be ready for a new install of python3.10.