This is an interesting one. I did try to update the drivers prior to coming here but didn't see anything on the HP site under that computer that seemed USB or eSATA related. However, the issue has seemed to clear up by the following:
Using device manager, I uninstalled the "device not found". Then, I rebooted the computer and plugged in a USB memory stick into the eSATAp port prior to restart. The device registered and was accessible. After leaving the device in for a couple days, I safely ejected the device and rebooted the computer.
So far (fingers crossed), everything seems to be working. It's been up for ~2 hours now without issue (and 2 days since I plugged in the USB device). I'm wondering if, at some point, a device was removed without ejecting it which caused the port to never really free up.