How can I improve clay soil?

I am not familiar with this but I would have said off the cuff that we need the drainage pipes with 1m - 1.5m distance from each other but that would probably be way too much, right?

I would start with the simple option and then observe if it is enough

I already started with drainage holes last year.

That didn't have a huge effect, when it rains you don't see any improvement, only afterwards the water seeps away in 1-2 days instead of 2-3 days.

I would only have drainage pipes installed once, it has to work right away but also not remove too much water. Just so that an excavator only has to come once.

One can also gain some control over water through garden architecture. A sunken garden collects some water in a hollow. It also looks nice because it can be planted with water-loving plants.

We will first observe how the soil behaves after we have mixed in a substantial amount of sand again

whether the waterlogging is still present or if it has already resolved itself.
We will shape the topsoil so that there is a slight slope towards the [Rigole]

I would add a few cubic meters of hummus/compost soil, which retains water but loosens the soil. Otherwise, with very clayey soil, it can happen that the soil dries out quickly, becomes rock hard, and rain after a long dry period infiltrates even worse.

He has already applied topsoil. The clayey soil is deeper down.
Make sure you loosen the soil with deep-rooting plants like foundation and soil organisms.
After the construction, water also stood here. In 2018, I had a lake on the play area after the rain. It resolved itself on its own.