You will have to experiment with each setting to see what works best. HP drivers are pretty good at giving you fairly decent color. You are limited in that it is a photoprinter, but I've gotten reasonable results using Deskjets and Photoprinters. Try selecting "Let printer decide color" instead of "let Illustrator decide Color". Also, sRGB may be OK for photographic printing, but try "Adobe RGB" instead with your Ai files. You may have a setting or 2 in Illustrator's Color Settings that may be affecting things, too. But try the aforementioned settings and see if the resulting prints are better or worse.
Your document profile says U.S. Web Coated (SWAP) v2 which should be used when you print to offset press, and you are printing to HP Photosmart B209a printer. Next to Document Profile it should show the profile of your printer. You assign the color space of your printer to your document in Illustrator by choosing Edit > Assign Profile and choosing the color profile of your printer which should come with the printer and installed with your printer software. You also have to know what color mode (CMYK or RGB) your printer can use. Alternatively, if you create documents for this printer most of the time, in the Color Settings for the Working Spaces, select the color profile of your printer and then every new document will be in the color space of your printer by default.
You don't have to do all this if you as John suggested in the print dialog, choose Let printer determine colors. This however will limit the creation of your colors in the U.S. Web Coated (SWAP) v2 color space while your printer may have wider color space so if you don't want to work directly in the color space of your printer, using wider color space like Adobe RGB is a good suggestion.