Is there any way to make a printer print if one of the color cartridges is empty?

In a nutshell, no.

HP (and to a lesser degree, Epson) made their printers too smart. (HPs wont even print if the cartridge manufacture date is too far in the past)

Cheaper printers (i.e. dumber printers) can be fooled. HP, not.

@elbanditoroso That is what I was afraid of? what if I put a black cartridge in the magenta slot or have they got that blocked also?

The printer will detect the color. If you look closely at the cartridge, there is a small computer chip, and that’s what identifies the type and color (and date)

Try to print using The Gimp.. It tends to not care about ink.

For example my moms HP printer/scanner won’t scan if it is out of ink. Scanning works fine with The Gimp.

If you are printing text you might want to give LibreOffice a go. Like the gimp I doubt they give a single fuck about how much HP makes from ink.

my moms HP printer/scanner won’t scan if it is out of ink

That’s borderline criminal.

I suspect the HP printer drivers are what’s keeping from printing when one cartridge is empty. If that’s the case then the GIMP won’t help. The GIMP is still going to be using the printing service that uses those drivers.

But I’d love to hear that is not the case… if it works please let us know !

I find giving the cartridge a good shake usually buys you a little time. Usually.