[SOLVED]No way to update python3.10 on my linux mint mate 20.3 Una

Here is how I solved the same issue in Ubuntu 20.04

Locate package libpython3.10-stdlib

$ apt-cache policy libpython3.10-stdlib
libpython3.10-stdlib:
  Installed: 3.10.4-1+focal1
  Candidate: 3.10.4-1+focal2
  Version table:
     3.10.4-1+focal2 500
        500 http://ppa.launchpad.net/deadsnakes/ppa/ubuntu focal/main amd64 Packages
 *** 3.10.4-1+focal1 100
        100 /var/lib/dpkg/status

$ ls -l /var/lib/dpkg/info | grep -i libpython3.10-stdlib
-rw-r--r-- 1 root root   14489 Apr 10 00:26 libpython3.10- stdlib:amd64.list
-rw-r--r-- 1 root root   23536 Mär 24 17:12 libpython3.10-stdlib:amd64.md5sums
-rwxr-xr-x 1 root root    1161 Mär 24 17:12 libpython3.10-stdlib:amd64.prerm

Remove manually the package

$ sudo mv /var/lib/dpkg/info/libpython3.10-stdlib:amd64.* /tmp

Repeat the same steps for libpython3.10-minimal. Then fix the broken packages

$ sudo apt --fix-broken install
Reading package lists... Done
Building dependency tree       
Reading state information... Done
Correcting dependencies... Done
The following additional packages will be installed:
  libpython3.10-minimal libpython3.10-stdlib
The following packages will be upgraded:
  libpython3.10-minimal libpython3.10-stdlib
2 upgraded, 0 newly installed, 0 to remove and 8 not upgraded.
5 not fully installed or removed.
Need to get 0 B/2.566 kB of archives.
After this operation, 68,6 kB disk space will be freed.
Do you want to continue? [Y/n] 
dpkg: warning: files list file for package 'libpython3.10-stdlib:amd64'    missing; assuming package has no files currently installed
dpkg: warning: files list file for package 'libpython3.10-minimal:amd64' missing; assuming package has no files currently installed
(Reading database ... 191581 files and directories currently installed.)
Preparing to unpack .../libpython3.10-stdlib_3.10.4-1+focal2_amd64.deb ...
Unpacking libpython3.10-stdlib:amd64 (3.10.4-1+focal2) over  (3.10.4-1+focal1) ...
Preparing to unpack .../libpython3.10-minimal_3.10.4-1+focal2_amd64.deb ...
Unpacking libpython3.10-minimal:amd64 (3.10.4-1+focal2) over (3.10.4-1+focal1) ...
Setting up libpython3.10-minimal:amd64 (3.10.4-1+focal2) ...
Setting up python3.10-minimal (3.10.4-1+focal2) ...
Setting up libpython3.10-stdlib:amd64 (3.10.4-1+focal2) ...
Setting up libpython3.10:amd64 (3.10.4-1+focal2) ...
Setting up python3.10 (3.10.4-1+focal2) ...
Setting up libpython3.10-dev:amd64 (3.10.4-1+focal2) ...
Setting up python3.10-dev (3.10.4-1+focal2) ...

Same thing happened to me. When apt fails I go for https://pkgs.org/, search the package, download and install manually. It usually works.