Looks like your digitizer has de-laminated from the screen. The digitizer is the glass that everyone breaks and calls the screen but the screen is actually the display that lays underneath the digitizer. The digitizer has the sensors in it that relay your touches to the motherboard. If I can find a diagram or something that explains it better I'll post that with this to make it clear.
Unfortunately, you cannot just press on this, and even if you were to lift the edge you would have to apply pressure from the screen side to reattach it to the glass/digitizer.
Did your phone get hot one day? It could have happened from flexing but chances are if this was a flex issue the screen/display would likely have been damaged. My guess is that it got hot laying on its back and the screen/display just fell away from the glass/digitizer. You can probably still find assemblies for this model. HTC was very popular so I would imagine you'd be able to find one still sitting on a shelf out there somewhere. If memory serves these phones don't have the best IP ratings so you could change this yourself if you're mechanically inclined at all. There are at least 6 videos on YouTube for this model, my guess without looking even. The hardest part will be that you have to open the phone from the back and take everything out to change the screen.
Speaking of YouTube you might also look to see if there is a remedy for this. I'm wondering if you couldn't just get it hot again and rest it on it's face and see if it heals itself. It's risky and I'm not saying to do or try this based on any experience, it just seems like it should work. Again, check other sources and assess your risk tolerance before taking any actions.
This isn't your phone, but it's representative of phone screens in general.
These are the steps to replace the screen for your device.