Partial upgrade error and system corruption

Recently I performed fresh installation of Xubuntu 20.4. During the process I however lost connection to the internet, so the installation was not updated. I do not remember exactly whatever the updates ever worked for awhile. But one day I wanted to check updates and I got this error: Failed to download repository information Check your Internet connection." I did a little a bit of research and I was able to fix it using these commands:

sudo apt clean
sudo apt update
sudo apt upgrade

From this point I am able to install updates from the terminal, but from the GUI it does not work very well (it does not offer all updates as it does with sudo apt upgrade). So since that I went through many rounds of system updating (both terminal and GUI offered further updates after installing the previous ones), I have also tried commands like sudo apt autoremove, sudo apt-get dist-upgrade to fix the issue (I did the installation for another user who is not very experienced and does not know how to use the terminal), but it did not help. Actually it got worse. So the system now is in this state:

1. During start up appears this screen bellow... The default kernel does not load.
2. and then these options appears. list of available kernels
3. During the boot process I have only grey screen instead of the Xubuntu logo. Just before displaying login screen UEFI logo appears for awhile. The logging mode of the booting process (Esc during the boot) works normally. The boot process takes longer than it used to before the system get corrupted.
4. Immediately after logging in I encounter this error message: error message about unspecified application
5. When I try run update via GUI this happens: partial upgrade error
6. When I clicked on Continue button during my last attempt to update the system from GUI, it offered me these updates... updates offered by the GUI update manager
7. however when I tried the same in the terminal, it offered me much more updates: updates offered by apt in the Terminal
8. After updating from GUI, the terminal offered me next, but the estimated size remained the same. That seems strange. (I deleted the image, because I would exceed the limit.)
9. And when I tried to update again via GUI after updating from terminal, it showed this: updates offered by the GUI update manager after updating from terminal
10. After rebooting I got new kernel version on the list, but the default one still does not work (the update in point 9 I declined).
11. When I tried update again from GUI - after skipping the partial upgrade error message, of course - it offered me upgrade to Xubuntu 22.04 GUI now offers update to Xubuntu 22.04

So now I need to remove the not working kernel images or at least that images which works set as default. Then I need to fix the problem with updates from GUI, the issue with system speed and stability and if possible the boot screen. But how I can fix these issues and from where I should start? Thank you.