Unable to add new printer setup app and need to update my macOS to Big Sur or later Versions

What Mac do you have?

If you go into About This Mac in the Apple () menu, you can find your Mac's model name, and hardware model year / sub-year. That will tell you how far you can upgrade the OS.

macOS Big Sur is compatible with these computers - Apple Support

macOS Monterey is compatible with these computers - Apple Support

macOS Ventura is compatible with these computers - Apple Support

macOS Sonoma is compatible with these computers - Apple Support

macOS 11 (Big Sur) is no longer one of "the most recent three" that get security updates, and the most "love" from application vendors – so if your Mac is eligible to upgrade to Monterey or later, you may want to do so.

How to download and install macOS - Apple Support

What year model is your iMac...?

see > Identify your iMac model - Apple Support

You need to have a 2015 or later iMac to run macOS 11 Big Sur.

see > Update macOS on Mac - Apple Support

RCHISE wrote:

imac 21.5in mid 2014

You can upgrade that iMac as far as Big Sur – but no further.

Note that 32-bit applications won't run on Catalina or Big Sur.

RCHISE wrote:

What is BigSur?

Version 11.* of macOS. Apple has a long history of using code names for Mac OS X / macOS releases.

with the bigsur update will the printer be able to be added?

I have no idea of how well HP's printer software will, or won't, work. But you can upgrade that Mac to macOS 11 (Big Sur). You cannot upgrade it to macOS 12 (Monterey), 13 (Ventura), or 14 (Sonoma).

If your printer supports AirPrint, you may be able to use it even without installing HP's software.

About AirPrint - Apple Support

Print wirelessly from your Mac to an AirPrint printer - Apple Support

Re: “How do I upgrade my computer?”

See the previously-mentioned Support article.