The error message is clear, the HP app is not compatible with your version of macOS (also indicated by the white circle with diagonal line through it). If this is an old printer, then HP may no longer provide macOS software support for it.
If your printer has an LCD menu panel, then usually you should be able to update the printer firmware that way.
Does you printer have an web interface for administration? That is another way to update the printer's firmware.
Be very careful updating HP printer firmware especially if you are using third party ink since many HP firmware updates will prevent you from using the printer with third party ink. If it is a multi-purpose printer with scanner, then you also have to be careful that the firmware update doesn't prevent you from scanning if one of the ink cartridges is empty. HP is known for tricks with their firmware updates. Personally I would avoid updating the HP printer firmware unless there is a critical reason to do so, but only after confirming that firmware revision will not negatively impact your printer with the points I just made (don't expect HP to actually tell you in the firmware changes list that they are going to limit your printer).