HP Support Assistant is not essential software. You already have WIndows Update to handle WIndows updates (both essential and optional ones). If HPSA is incapable of updating itself without errors, just go to "Add or remove programs" (or "Apps and features") in your Windows Settings/Control Panel, then select HP Support Assistant, and select "Uninstall." Don't worry about finding and installing the updated version. You will be fine without it.
Updates to HP device drivers and other such things are much less frequent than Windows system updates. Your PC should be fine with just Windows Update. If you have an HP printer or all-in-one, visit the HP Support website once a quarter or so and look for new drivers or firmware for your model. Also do this if you run into trouble that a reboot can't fix. Don't worry about software updates other than drivers and firmware. You don't need them.