Do software updates searches include printer drivers

I think you need to update driver by yourself.

http://www.hpdriver.net/hp-deskjet-f380-driver/

Your printer is not listed here: OS X: Printer and scanner software available for download - Apple Support so you'll have to download updates from the website of the printer manufacturer.

Hello again!

You might not have HP Device Manager since it's not default isntalled on a Mac

Instead, if you don't have any other printers or scanners installed do this:

1. Boot your Mac to Safe Mode: OS X: What is Safe Boot, Safe Mode? - Apple Support
2. Open the Finder and click "go - go to folder"
3. Enter: /library/printers, click [go] and delete content of that folder
4. then, go > go to folder and enter: ~/library/printers (including the ~) and delete the content of that. It's possible this folder doesn't exist.
5. lastly, click go > applications, and check your applications folder for HP software and delete it. Also open the subfolder Utilities, check for HP software and delete that as well
6. reboot your OS X normally and your printer software is uninstalled