Metal plug blocked in sink

Hi,

any advice would be urgently appreciated. Sorry about my lack of plumbing vocabulary or knowledge but I've got one of these mechanical(?) metal plugs which you bring up and down by pushing a lever(?) down or pulling it up just behind the taps and the plug has become blocked and won't come out. I think the bottom of it has fallen off inside the pipe so it won't work anymore. I simply want to know if there's any way I can get the plug out by attaching something to the top and trying to pull it out that way as I can't get a knife or a screwdriver to it. Would a plunger work at all in this situation? I've managed to get all the water out of the sink.

Thanks

Take the trap off

that way you can push the plug out with a screwdriver and recover the bits that fell off at the same time.