Hi Olga. You're absolutely right, the leeks in the photo look miserable.
I will tell you the possibility that this situation may have its origin, and the solution is in your hands. The “grass” plant you refer to is a postillon (Arum maculatum) but there is also the sapelar (Stellaria media) and I would say that there are also some carrots (Daucus carota). And yes, the leek likes the art of grass, because he's a humid lover. The most important thing is to have a deep, fresh land, and a lot of manure, with manure or old pit. As I see in the photo, the ground is regular. The emergence of Ilarrondoa also indicates that the land has not been prepared in depth. And finally, to have a couple of months, despite the circumstances, they're not that bad ... Next time, if possible, change by the end of June or early July.