Bathroom sink not draining

[Photogemic]

My bathroom sink is not draining and I've tried Draino. After watching some youtube videos i want to try and take out the plug/stopper. We have one of those where you plug a lever behind the tap to raise and lower it into the plug hole. All the videos show just a push in one and how to remove it (and to get out all the disgusting gunk) Anyone any idea how to do this on my style plug? Just trying to avoid a plumber and the extra cost.

Thanks!
Gx

Does the head of the plug not unscrew to take it off?

Can you remove the U-bend? It's probably full of gunk

Put a washing up bowl under the U bend and unscrew that. Probably blocked with hair and grit. I always keep a wire clothes hanger for plug draining. Stretch it out full length and use a plier to bend a smal hook. Like unblocking a vacuum cleaner tube.