I had a black only HP printer and it was a pain. It only worked well when I changed to Calidad brand refills. The genuine ink blocked up every time I used it.
After soaking you can try blowing into the vent holes on the top. Dab a tissue on the nozzles and if the ink flow is not even, cover the free flowing parts with your finger while you blow.
I used to moisten the nozzles with metho which was much more effective than water. If you havn't used it for months I wold go straight to the metho.
Small HP inkjets are crap and very expensive to run since the heads are integral with the cartridge. This "feature" is just plain stupid. I had a much better run from a Canon but you would be much better off buying a laser printer which is far cheaper to run.