When you see prints that are darker than your on-screen display that frequently means that your monitor brightness/luminance settings are too high. An automatic calibration doesn't alway catch that, so that's something to check out.
I'm not familiar with that particular printer, but HP photo printers do have ICC profiles included with the drivers so you can work with a color-managed system. The driver should also have an option to let the application manage colors -- when you are in Photoshop try the View->Proof Setup->Custom dialog, and in the dialog open the Device to Simulate list. Toward the bottom should be a list of monitors and printers with ICC profiles for a combination of printer, paper and inks -- does your printer show up?
If not, you might go onto the HP Web site and look for drivers for that printer and download everything you can find. Bear in mind, though, that these profiles are for printers that are designed for photo printing. "Generic" printers will not likely have ICC profiles.
If the profiles are there, you could run tests -- try a print letting the printer manage the colors (make sure Photoshop/Lightroom turns off its color management and the printer has its own scheme in the Preferences/Color Management tab) and then having the app manage the colors. You have to explicitely set this both in PS/Lightroom and in the printer Preferences/Color driver utility.
Aso, if printing out of Photoshop, you can use the View->Proof Setup utility to proof you print, that is, to see what Photoshop "thinks" a print will look like, but that will only work if the ICC profiles are there.
I have two HP photo printers, and they do OK as long as I folow the right process. But I only print with them using HP papers and inks -- I haven't experimented with third party papers or inks to compare results. If you can find a third-party paper with HP printer profiles, then you should be OK.