I don’t know about “chalk” but vinegar dissolves most hard water setements.

Fill a plastic baggy and tie it around the faucet so it’s summerged in vinegar. Use like an old toothbrush on it every couple hourswhile letting it soak for a while and it will all come off.

Now that that’s gone, try the old methods to remove the plastic piece

If you still can’t you have 3 options:

Demolish it, it’s plastic and the parts staying are metal, rip it to shreds with pliers.

Use a strong adhesive like super glue to glue something to it, wait for it to dry then unscrew it.

Call a plumber

You could also just replace the whole faucet, at a certain point that’s just easier.

Its broken anyway, I’d jam a slot screwdriver into the plastic and see where that gets me, either you get it screwed or out or you break it into pieces.

I had this same situation happen. I was able to use a screwdriver and jam it in and make a little notch. Pushed the screwdriver around until I came out. It took a WHILE though.

Take the whole faucet off. If there’s any good way to grab it you’re probably gonna be able to get a better look at it with it sitting on a table or in a vise and have more room to get tools at it.

If, like most of the sinks in my house, those shitty plastic basin nuts holding your faucet to the sink are stubborn and almost impossible to get off and you don’t have any tools that will do the tick- those nuts are maybe a few bucks at any hardware store. I tend to just take a sharp chisel to them with a couple good taps from a hammer and cut them off.

Give it a good soak in a bucket of vinegar or CLR to clean as much of the lime and crud off as you can. Trying to get a baggie full of it attached to it in-situ like a lot of people recommend is usually a pain in the ass (I usually prefer to soak a paper towel in it to wrap around it to soak for a while, easier and it probably uses less vinegar/clr.)

Like others have said, you can try cutting a slot or something into it to get it out with a screwdriver. If it’s plastic or cheap metal you might be able to just break it up and take it out piece by piece.

And if all else fails, you have your faucet off and unless you’re really particular, faucets aren’t that expensive. Just replace the whole damn thing.

Could something like a potato cut in half work? It’s hard to tell how deep the broken pieces are in there. It would grip on the broken parts and then hopefully when you turn it then it moves. Maybe after a soak in vinegar as another suggested.