HP 1610xi printer printing very lightly, barely legible, after changing to new HP cartridges and doing cleaning maintenance repeatedly.

I guess I should ask if you used an HP replacement cartridge or another brand? The printer may not recognize a 3rd party replacement cartridge. Next, assuming you are using an HP cartridge, is it new or a refill? A refilled cartridge may have dried ink blocking the ejection ports on the cartridge.
So I would start with a reset of the printer. remove the ink cartridges. Disconnect the cables, and press the power button for 30 seconds. Then reconnect the cables, turn the unit on and reinsert the ink cartridges. Do a test print. If the black still doesn't work try cleaning the print head.
Click the setup icon and go to Printer Maintenance, click Clean Print Head. You should hear the machine doing this. Print a test page and see if this fixes the issue. If it doesn't or only partially fixes the problem wait 10 minutes and clean the print head again. Hopefully, this will at least improve the issue. If the print quality still doesn't look right wait 10 minutes and repeat the cleaning again.
Hopefully your issue is fixed. If it still isn't printing, could it be your paper? Is it the same paper you were using before? Some kinds of paper do not absorb ink very well.
Hope this helps.

1- Open the printer
2- Remove the ink cartridges from the print head.
3- Use cotton swabs dipped in hot water to thoroughly clean the print head.
4- Repeat this until there is no ink residue coming off on the cotton swab.
5- Once finished, allow to dry and replace the cartridges

Get the 64 bit driver at Kodak Help Center.

If any more problem encounter,post or thumbs back.

Depending on the age of the printer, the service staion may need to be cleaned. The service station is where the cartridges "park" when they are not in use. Over time ink can collect in this area and therefore the ink cartridges do not get a thorough cleaning. I would go to HP's website and look for cleaning instructions for your printer model. Hope this helps

Check nozzle and do head cleaning.

Would help knowing the make and model of the printer but sounds like a drum fault to me.

Sounds like your pump station has gone bad. Now with two new cartridges loaded you have flooded the well and it's leaking. Clean up the mess and try using hot water in the plastic tubes to unclog them. If that doesn't work junk it and buy a new printer. It's not worth paying a HP service center to repair it as the cost will probable be more than half of what you paid for it. Sorry.

Are you using compatible cartridges? If so have you used them before?

Make sure that you run a cleaning cycle in the printer and then print out a test sheet to see if there any issues. You may have to run the cleaning proces 2 or 3 times to get it working properly.

Here is the HP support site, for manuals downloads etc.
http://h10025.www1.hp.com/ewfrf/wc/product?product=466210&lc=en&cc=uk&dlc=en&submit.y=8&submit.x=3&lang=en&cc=uk

Your ink cartridge (s) is dried up, even some new ones are dried up, even tho' they're sealed.

Some self-tests don't use black.