What's the problem?
The message "Check your Internet connection." is unhelpful. The application can (and does) check your Internet connection and, obviously, (since the update ultimately works) your Internet connection is fine. This message is a low-effort way of saying that they ran into some, unspecified, problem reading from one or more of the configured repositories. But, as you've seen, it does not necessarily mean that the download failed completely.
Using other tools, such as the 'Synaptic' graphical package management tool (available in 'Software Boutique') you can get more information. For example, with Synaptic, I currently get the following response when attempting a 'Reload' (Note: A similar, but less descriptive, message is available when updating from Software Boutique itself):
So it seems that my system (Ubuntu MATE 22.04) has a reference to archives from the now-defunct [21.10] 'impish' release. But even the above message confounds the problem since it's saying "does not have a release file" when, in fact, that repository does not exist at all (due to rollout of the EOL).
The 'Software & Updates' dialog (accessible by clicking 'Settings...' in the Software Updater) is where repository locations are configured. In my example, this includes the configured reference to the defunct 'impish-security' distribution at http://security.ubuntu.com/ubuntu. [It's interesting that this isn't using 'https'.]
So what's the fix?
If your case is similar or analogous to my example, the quick fix is to disable the offending software source that is listed in the 'Other Software' tab (shown above). Once this is done, Software Updater will stop complaining about this.
Alternatively, if you happen to know of a new (and still secure) location of the repository, you could change the URI (and/or Distribution) to specify that.
What are the risks?
Disabling software sources prevents any associated security patches from being applied, so this should be done with an awareness of why the sources were present in the first place. Ideally, if you are deleting a software source, you would also uninstall any packages that it provides.
In my case, for now, I don't know why there is a source called "Important security updates" that points to an older version of Ubuntu. Maybe it was added by some component that I installed from a third-party but, without retracing my steps, it's hard to know. What I do know is that this source is no longer providing those "Important security updates", so ... oops.
