Thanks for the clear details of your setup. To clarify, are you using a standard HDMI cable? Or perhaps are you using a DisplayPort to HDMI cable?
Pending this clarification, let’s investigate further into your Windows Display settings to see if the non-working HP monitor is at least being recognized by your system. To do this, please keep both displays connected to the dock, then follow these steps:
Right-click on any empty area on your Desktop, then choose Display settings
Under Select and rearrange displays, do you see two labeled boxes or one?
Since the DVI connection on the dock is working fine, it’s likely the case that the DisplayLink driver has installed and is functioning properly (otherwise, both displays would not work). If you have one available, the first steps we’ll want to take are to swap out the HDMI cable with a known-good cable to rule it out as a potential culprit.
If this isn’t an option, you could also try connecting the HDMI cable directly to the HDMI port on your laptop and seeing if it displays without issue.