Graham,
I want to emphasize what Bill has said. I have been using Epson printers since about 2000. At some point, somebody gave me some HP paper and it did not work well. I researched the issue and learned that the difference in technology was such that HP paper did not work well with an Epson printer. The reverse would be the case as well. I mostly only use Epson paper. That is the easy way for me. The paper comes with profile instructions. Since you are using an HP printer I would recommend you mostly stick with HP paper. In my experience, the premium paper vs. the ordinary photo paper makes a big difference. Generally, you can download profiles off the web for the papers you use. That is at least true for papers that are used with an Epson printer. I assume the same applies to HP printers. It is important to let your Adobe software control the colors, not the printer. I only print from Photoshop so I do not know if the printer interface is the same for LR and Photoshop. I use a Mac when I print. There are two places to put in profiles. One is the printer profile and then the media type. If I do not make sure both are correct for the specific paper I use, the results are not satisfactory. I have not normally profiled my papers myself. Several years ago Eric Jones did some tests and at least for Epson, he found the supplied profiles were almost exactly the same as what he measured. Like Bill has indicated when I print B&W images I send "full color" images to the printer. That means the R, G, and B channels are identical.
Another thing I do that apparently a lot of professional people use to get consistent prints is to use the info pallet in Photoshop. It certainly makes sense to calibrate your monitor. However, your eyes built-in auto white balance can mess you up and things may look like they have a color cast when they do not or vice versa. So, I think it is wise to calibrate your monitor and also to examine the numbers in the info pallet. Neutrals are going to be pixels where R=G=B. Personally, in Photoshop I also depend on seeing the numbers for the colors in LAB color mode. For me they make more sense than the RGB numbers.
BTW, my Spyder 5 does have the capability of profiling papers. I will add that I have boxes of paper that I have had for a long time because I was not happy with the results of using them.
Jim Gray