Currently on Lubuntu 20.04.2, but being repeatedly asked to 'upgrade'

I don’t know your issue, so I’m only guessing.

I would open a terminal and

sudo apt update

to ensure your package lists are all updated, and look for any warnings & errors. Assuming that’s good; I would

sudo apt full-upgrade

to apply all upgrades, again watching for anything that looks unexpected/wrong…

If you only sudo apt upgrade for example; some packages may not be upgraded; causing notification of upgrades available to re-appear (until you apply them with full-upgrade) but as stated I’m guessing.

FYI: I’m on the development release, and do all my upgrading via terminal so I’m aware of packages that are upgraded… so I’m a little rusty on GUI tools on upgrading, thus my suggestion of commands.