Yep, that can be frustrating.
Some of the in-place upgrades from Win 7 to 10 go poorly.
It may be that it still has Win 7 drivers lying around for the printer.
How about some specifics, like which HP printer it is, and how connected - eg USB or WiFi or Ethernet ?
I have seen Windows install a PCL6 driver when the printer is only PCL5 capable, so you may have to find the correct vintage driver from HP.
Does it show up in the Device Manager?
Any errors in the Device Manager?
Here's what has worked for me:
1. Run Windows Update several times. Reboot after it installs updates, and check again. The update process does not always present every update when you would expect it to.
2. Disconnect the printer from the computer and power it off.
3. In Device Manager, View... Show Hidden Devices.
This will reveal "phantom" devices.
4. Delete all instances of the printer.
On the last instance, it will likely have a message asking if you also want to Delete the driver. Check that box.
5. Reboot the PC.
6. Install the correct HP software for the printer, downloaded from HP.
If you can't find it, try this for PCL5 - you may have to dig around:
HP Universal Print Driver for Windows - PCL 5 Software and Driver Downloads | HP® Customer Support
or for PCL6:
Universal Print Driver | HP® Official Site
and also:
HP Universal Print Driver Series for Windows Software and Driver Downloads | HP® Customer Support
Part way through the install, it will likely ask you to plug in the printer.
Follow the steps and you should be good.