Next steps for the new lawn

Here is a current picture. We also had the problem that a few days after sowing, we had a heavy downpour and therefore some areas were washed out.

Where it is particularly bare, proceed as follows. Take potting soil, mix it with grass seed, plenty of it, and some fertilizer. Scatter this where there is hardly any thin grass. Water, wait.

Where it is especially bare, proceed as follows. Take potting soil, mix it with grass seed, plenty of it, and some fertilizer. Scatter this where there is barely any grass. Water, wait.

And when is the best time for this? Now three weeks after the first sowing or after the first mowing?

That doesn't matter. I would mow, then overseed. But it certainly also works the other way around. The important thing is that the soil must be kept moist.

Can someone possibly tell me what kind of weed this is?

Thank you