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.