Tony, Welcome to RetouchPRO!
See my video tutorial on
Color Shift. This may be of help you understand what the printer wants from you
Every printer is different. Same for monitors. There are a couple things you need to be sure you do. I use an Epson, but the theory is the same.
1. Use the Print with Preview option in Photoshop.
2. Set the Color options as shown in example 1 below. RGB and Relative Chromatic.
Now you need to study your printer driver dialog box when you choose PRINT and select your printer. The terms may be different, but below is how I set up my Epson for printing.
1 Media Type: VERY IMPORTANT!! Depending on the kind of paper you are printing on, you need to set this correctly. For optimum results you need to use a photo glossy paper. Expensive, yes, but print on plain paper for drafts first to check things out.
2, Quality: I always set for high quality and high quality halftoning even when printing on plain paper.
3. Finest detail: I also request this even when printing a draft.
My Epson has several other print option screens, but this dialog box is the most important. Options #2 & #3 above use more ink, but the results are more consistant.
Hopefully this info along with my Color Shift tutorial will help you understand what your HP expects from you. Always send RGB files to the printer and let your native printer driver handle the CMYK conversion.
I've never calibrated my monitor and I do prepress work. I know that's probably not the "correct" way. But when I'm working on stuff that is going out to a 4 color process image setter, I use my vendor's ICC profiles for the CMYK conversion and I've had excellent results.