And what will happen if you press "Y" after this error message?
Also please note that this is a method for Zorin 16, not for 15.3
I don't press "Y" because it says this:
Auto installation is not supported for 'unknown' distro so all dependencies may not be installed.
Please install manually as mentioned in 'http://hplipopensource.com/hplip-web/install/manual/index.html' web-site
Given that this is my second day on Zorin, and in fact Linux, I won't be pressing anything that suggests "all dependencies may not be installed" because I have no idea what that means, nor the consequences. It says "Please install manually...." but the problem with that is HP's manual installation instructions are for Ubuntu 8.04 and 10.04. I believe Zorin 16 is based on 20.04?
You'll note that I said this:
It is correct.
I missed that one. Thanks for pointing it out.
You could also try to install HPLIP simply from terminal.
https://ubuntu.pkgs.org/20.04/ubuntu-main-amd64/hplip_3.20.3+dfsg0-2_amd64.deb.html
sudo apt-get install hplip
I tried what you suggested and got this:
What do you reckon, should I go for option 0 or for option 0? 
Why does it say the newest version is 3.20.3 when in fact it is 3.21.6 and I have that saved in my Downloads folder.
For me, it looks like you already have HPLIP installed.
Did you login/out or reboot to enable it?
No, do I need to? I can see the HP Envy 5540 printer, I added it it yesterday, but when I go to the settings there is nothing in there about scanning, it's just set up as a printer.
Not always but some drivers require reboot and HPLIP is one of the latter.
I've rebooted today already, the printer was added yesterday. There just seems to be no options about scanning.
To use scanner, you need to start Simple scan (can be installed in Software) and set up scanner in it.
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Simple scan isn't there.....but....Document Scanner is there. I just launched it and it detected my HP Envy. So that will probably do it I suspect.
Thanks for your help with this FrenchPress, I am a total newbie to Zorin/Linux so don't even know where to look for most things! I'm glad there are helpful folk around like you. So thanks, I appreciate it.
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Glad to here that all seems working now 
If you think the problem is solved, you can tick 
to mark the thread as solved.
This will help the future visitors to this forum to find a solution they are looking for.
PS
Document scanner is yet another name for Simple scanner.
OK...just point me towards where the tick is....
There are three dots in your original posting.
By clicking it will show you a option of marking the thread as solved.
I don't see that when I click the three dots, but then I didn't start the thread.
You are absolutely right about that.
Well at least you can tick 
in the solution I gave you 
Done
Future reference: For "unknown distro" on HPLIP, you need only go into the files and change "ubuntu" to "zorin" and then run the installer - it will work just fine.
Sorry but I don't understand what you mean. I just checked the hplip folder in my downloads folder and I can't see 'ubuntu' anywhere in the files or folders.
It is one of the files within the installer- and is content within the file. It is not a file named Ubuntu and I uhhh... realize now I cannot remember the name of the file.
It's been a while... And I commented off hand as I thought your issue was solved.
I did a search of posts on the topic of the forum as we on here have covered this a few times... Haven't found that particular technique but I did find a post from azorin re: Zorin 15, but should still be applicable: