Looking at the summaries of reviews on Google, plus looking at the CNET review of the printer it looks like it is designed as a fully multifunction office printer. The strengths all seemed to be centered around office use, what with the document feeder for the scanner, built in fax capability and even full duplex printing. Print quality for documents and "graphics" seems to be adequate too. To get all of that into a sub $100 US printer though it means that they will have made compromises in other areas. About the only place left for those compromises is photo printing. While it's possible to print photos to glossy photo paper I'm pretty sure that as TeamSpeed alluded to it is only using the CMY inks for photo printing. This means that it has to mix the three to make black, and that really hasn't worked for photo printing for something close to twenty five years.
The problem with mixing CMY to create black is that when you mix them up what you actually get is something that is closer to a really really dark green/brown colour. Compared to actual black ink though it is quite a bit lighter, and of course has that colour cast. That is exactly what you get in the result that you have posted. Initially I didn't consider it as a factor, since I didn't actually go away and look at the exact type of printer you have.
There are some relatively cheap priced multifunction printers that do actually do well in the photo printing area. Canon have some in their Pixma line of printers, I have a rather old MG5150 that actually does very well when it comes to printing photos. It's less suited to a full on office environment though, since it lacks the feeder for the scanner, and has no fax. It was also a little more expensive in that it cost about £120, or about $150 - $175 US at the time I bought it. The big advantage that it has for photo printing is that it uses the three CMY inks, and also a photo black ink, as well as a pigment based black for document printing. This additional black ink allows for a much better looking photo, as it is able to create the blacks, not just the mixture of the three CMY inks.
Unfortunately I don't think there is going to be anything you can do to make that printer produce better photo output. For this sort of printer I would normally suggest that you use an OEM branded photo paper and of course OEM inks to get the best possible quality. That way you can select exactly compatible settings in the print driver for the paper, and you also have the correct ink. This would allow the printer to get a reasonable colour match regardless of the software that was being used. If your printer driver offers you choices of a specific brand, and version of photo paper, rather than just the generic Glossy/Matt choice then I would try that particular paper in it, since it should be optimised for that paper. What I would not do is use a different premium brand paper, such as the Canon Platinum Pro, which is at the top end of cost. I just don't see you getting any improvement from it.
I do actually use Platinum Pro, along with OEM ink in my Pixma printer, and they do give excellent results. But only because I have the required profiles for that combination of paper and ink. Without them, and just using a generic glossy photo paper setting I wouldn't expect any significantly better result than any other reasonable quality glossy photo paper. I would put your HP paper into that class in this case.
When it comes to printing photos I have used either Photoshop, or more recently Lightroom as my software of choice. Since I have shot pretty much everything in RAW since about three days after getting my first DSLR back in 2005, I generally find that I have to use some program or another to work with and edit the images. Also since for most common photo paper sizes other than 6×4 you have to crop the image to some extent to get it to fit on the paper that requires more than just using the system print functions. Another advantage of using programs like Ps and Lr is that they can very often offer you more control than you will get from using the printer driver on it's own. Although it is a potentially complicated process for the novice using these programs, with their colour management systems you can get a much closer representation on the computer screen of what the printed image will finally look like. The difficulty is that to make it work for you you need to have a reasonable understanding of how different types of devices, such as screens and printers, produce the colours, and how you have to compensate for those differences between the devices.