Yes, you should be able to get a profile for your printer. Depending on what paper you use, the manufacturer may offer this free of charge (PermaJet do, buy their paper, print the target and send the print to them, they then send the profile back).
How did you calibrate your monitor? Some gadgets have simple add-ons that will calibrate the printer too.
And something to think if... How are you printing? And what version of Photoshop are you using? Overly dark prints are usually caused by double colour management - if you say 'Photoshop manages colour', not all HP's turn off the printer side colour management so you're effectively managing it twice which usually means dark prints... Try test prints using 'Photoshop manages colour' and 'Printer manages colour' and compare the results. Also, look in the printer dialogue options and see if you can turn off printer colour management - and then use 'Photoshop manages colour' and see what happens.
Hope that makes sense!
Cheers
Andy