Reddit mentions of ITW Brands 11353 6PK #30 Dry Anchor

I'm from the US so we may be using different terminology but the plastic thing that you put into the wall is what we call a "wall anchor". I think they are what you call "plugs".

If you look at the package of wall anchors you've purchased it says it's for use in "solid brickwork, masonry, concrete or blockwork". Those type of wall anchors are not intended to expand very much and will not grip drywall well. You need dry wall anchors something like this. Also pay attention to the the amount of weight they are rated for, something like this the higher the better. If you there are studs in that recess that's the best thing to do is screw into the studs directly using some strong screws and forget the drywall anchors.

Otherwise get drywall anchors and unless the anchors are quite a bit bigger than the holes you already made you should not reuse the same holes.

Good luck and also keep in mind walls in a home are not always exactly 90 degrees so you may need the boards / ledges that the top will rest on to be a bit bigger or you can cut the top to custom fit the shape of the recess.