Your first problem is you have a HP. I would not have a HP given to me for a gift. I have Epsons that have separate ink cartridges for each color.
Second, there is really no way to scrimp on ink. They sell you the printer for pennies then charge you an arm and a leg for ink. They ALL do it. I spent years testing printers/ink/3d party ink and came up with this.

1. Get Epson
2. Get separate cartridge ability.
3. Find out which epson can handle the "HIGH CAPACITY" ink cartridges that some Epsons take.
4. Don't mess with 3d party ink. The quality is not there.
My R380 takes high capacity carts and, while the initial cost is high, I can print for half the year on a set.

Now, you might look at this idea. If you are going to print 50 4x6's you just took of the kids at sea world then outsource it. Costco is good. AND a lot cheaper. My printing is specialty printing of 1 or 2 that just got Photoshopped and means a lot to someone. For example, my Granddaughter just got on the school basketball team. I took 78 pics of her playing, found one that was exceptional and printed an 8x10 for her to put into a frame. The rest, they have on a CD so they can view them any time. If they want a print of 1 or 2 of them, I'll do it. The rest, my Costco has prints for 18 cents if they want to go that way.
My long-winded 2 cents.

Canon Selphy CP900. Prints 4X6 and is very small, and sell for about $80.
No ink problems with drying or spilling because it uses dye sublimation. The "ink" is a roll of film that has yellow, red and blue dyes that are transferred to the paper (glossy only) by heat.

The ink/dye packets are not too costy. It costs me about 27 cents per print. If you're on site without power, there is a battery system (Canon adapter with Wasabi battery instead of Canon), which will cost more than the printer, but worth the cost.

You can also take along a laptop, download pictures from camera to laptop for quick editing and then to printer.

You can probably tell by now, I like mine! :)

EDIT: The final step in printing lays down a clear UV protective layer, making them archival, waterproof and damn near impossible to tear. Takes about one minute per print.

I used to have a Kodak that was the same...gave it to my Dad years ago along with a Kodak camera that docked on it. It was perfect for his level (lack) of tech expertise and the pictures were great. I have a brand new one that I bought on clearance shortly thereafter because I missed it more than I thought I would, but never set it up as I was in the process of moving when I bought it and then I bought my big HP (6 cartridge) and it does such beautiful prints I never needed the small one. I'd forgotten all about it until I saw your post so I went and looked...yep, it's still in the closet in the office. Checked on Amazon for cartridge/paper refills...they are twice the price they were when I bought it! I wonder if it's because it's Kodak...

The Kodak is able to print both glossy and mat. It is limited to 8x10, but reasonable quality prints and easy to use. I had one for a while, but the 10 cartridge Canon I now use is more versatile in sizes and textures. It requires a higher skill level to get the most out of it, and it is not cheap to use. Just the same, for prints up to 13x19 it is the best I've seen. For larger, I switch to Epson. The latter needs a dedicated room though.
Without question, for large numbers of ordinary quality prints, any of several on-line and discount store printers are far cheaper than anything we can do at home. For just a few, it is simpler just to print them at home and bear the small extra cost.