I don't have expertise in matching printers with "off-brand" papers, you could benefit from some of that...
A couple things that you can try, though:
First off, consumer monitors are typically way too bright out of the box. When I get a new monitor (or laptop) I first off turn the Brightness way down, until the display almost turns ugly. Try that, in fact see if you can turn it down far enough to match your prints!
From there, monitor calibration can help, although I've found it's typically more effective with colors than with monitor brightness/print darkness, which is why I recommend cranking that brightness down right off the bat!
Then, I'd advise you to experiment with the Soft Proof tool in Photoshop, since you say you're using PS. What I don't know is what comes with the driver/software for your Canon printer. If the software provides "profiles" for Canon paper, printer and inks, then the first thing is to pick a profile for the Canon glossy paper -- hopefully it will do a decent job of matching the results of the HP glossy paper, although like I said, I have no expertise there.
In the Soft Proof tool, try that profile, and see how it lines up with your prints! There may be some "tricks" to using the Soft Proof tool, maybe others can chime in there.
While in the Soft Proof tool you can do "tweaks" in the image editor, and then do test prints tweak and test, and see if you can get good results in the Soft Proofing. If you do, you can save them by creating an Action, because for one thing you will likely want to save an image for Web use or "sharing" before tweaking it for printing, so keep an eye on what you are doing so you can record it!
One last thing: when viewing prints to compare them to an image on your monitor, you should realize you are dealing with two ways of illuminating the image. The monitor "back-lights" an image, whereas with a print you are seeing a reflection from available light. So it's typical to view a print using generic indoor lighting, which is not good for comparing with the image from a monitor! Because of that it's best to view prints in "good" neutral light, light that is similar to viewing them in daylight (not direct sun, though). In fact, people who are serious/professional about printing have lights that are by design good for brightness and neutral for color for the purpose of viewing prints. And, a lot of those folks also have their monitors "hooded" to prevent the ambient light from interfering with their processing work!