I think what John meant was if you lift it up and maybe put a slotted screw driver into the half moon, the whole piece should come out. If it doesn't come out then it my be stripped or corroded on. If you get really desperate you could try going underneath and prying, I wouldn't go too crazy cause it may disturb the putty in the drain and cause a leak. It's kinda hard to diagnose some stuff online, even though its something as simple as this, sometimes you need to physically be there to be able to do it and try different methods based on what you see. But give those ideas a shot and see how it goes, best of luck!
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