All HP cartridges, the Canon BC-02 ,Lexmark and Dell cartridges have the head built into the cartridge. When these cartridges become empty the ink trapped in the micro jets is exposed to air from the inside and the outside. The ink begins to dry both ways and when dry, forms a slug of solid material in the micro jet. When clogged, the ink will not flow out the jet. What you see on the paper is skipping, banding or no print at all. If you fill a clogged cartridge, the new ink will not unclog the head. OK, how do you unclog them?
1) Hold the printhead under a stream of hot water for about 10 seconds - dry with a soft cloth - reinstall in printer and print this page a few times. (for Color) or  (for Black)
2) If that doesn't dissolve the clog. Fill a saucer with half hot water and half ammonia - place the cartridge in the solution for about 2 hours - dry with a soft cloth - reinstall in printer and print this page a few times. (for Color) or  (for Black)
Here is what the professionals do. They use a temperature controlled pan (like a fry pan) and set the temperature to 180 degrees F. A small amount of the ammonia and water solution is put in the pan to cover the bottom. When it has reached 180, the cartridge is put in so that only the head is under the water. The cartridges are cooked for about 30 minutes. After the 30 minutes, a puff of compressed air is applied to the vent hole (you can use a syringe or squeeze bottle). Then while the cartridge is still warm, it is installed into the printer and the cleaning cycle is run several times.
I know, who wants to waste a $50 fry pan to unclog a $20 cartridge. The inks will wash off with hot soap and water and a little bleach. If you want a simpler approach, use the saucer with some hot water and ammonia as I mentioned before. This may not be as effective due to the lack of temperature control and cook time, but it is worth a try.
3) Here is what one customer wrote back and said, "But what really worked was the suggestion from my dad when he saw me doing this. He suggested steam. So I heated up the teapot. When it began boiling I slowed it down to a steady steam. I put the head near the pouring spout (with a stainless steel bowl underneath). The head began to weep ink. Now the cartridge prints fine!"
4) Here is what another customer wrote back and said, "Here's an easy way I found to clear my HP ink cartridge. After I had re-filled it, when I put it back in the printer nothing came out -- it was completely blocked -- I guess because I exposed the ink head to air for too long as I re-filled it. Well, from your hints, I started by putting the cartridge, ink head down, in a metal cup into which I had put small amount of 1/2 boiling water and 1/2 ammonia. Nothing happened. So, I replaced the tin cup (in which the water cooled very quickly) with a ceramic cup, put 1/2 boiling water and 1/2 ammonia again (still topping just above the ink head), adjusted the cartridge so it was back against the wall of the cup, so it would stay level, and placed the ceramic cup in a small, 4 cup, metal cooking pot into which I surrounded the cup with boiling water. This way I was able to keep the water surrounding the cartridge always hot, by frequently pouring out the water in the pot with fresh boiling water from my tea kettle, which I kept re-heating. Not long after that I noticed that the water around the cartridge started turning black -- the blockage on the cartridge ink head was cleared -- the ink was coming out freely and fast. I quickly took out the cartridge, placed a folded paper towel over the ink head and as quickly as I could replaced the cartridge in my HP printer and printed the pattern you have for black ink at 
And, guess what: the solid black rectangles in your pattern came out solid black, and a page of .doc text printed perfectly. It is such a simple solution to an apparently complex problem, and saves you from buying a new HP ink cartridge (which aren't cheap!)."
This is about all that can be done. HP cartridges are the worst ones, because their ink dries very fast. All of this can be avoided if you do not let them go completely empty and if you store them in a sealed zip lock bag with a damp paper towel or wrap the cartridge in Syran Wrap (plastic sandwich wrap) this will put a piece of plastic next to the printhead and prevents air from getting to it. Old empty cartridges that have been in a desk drawer or in a hot storage place for days are NOT candidates for refilling.
Get in the habit of topping off your cartridges and try to never let them go empty.