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Edit: You could also play around with the angle in the auto smooth setting, but your screenshot kind of looks like there is a different issue. Hard to tell from the picture alone, though.
Did you try using the Bevel Modifier with Bevel Weights as the Limit Factor? That usually gives alot of nondestructive control.
As in you can change the Ammount of Bevel Segments aswell as their radius etc. just by dragging sliders and assigning weights.
On a side note, for the Bevel Modifier to work properly, the Scale of the Object has to be uniform, ideally 1/1/1.
Here's the blend file: https://drive.google.com/file/d/1KxFvz03wagXdBVovx4qz9qxqIkjbJ1I4/view?usp=sharing
https://drive.google.com/file/d/1KxFvz03wagXdBVovx4qz9qxqIkjbJ1I4/view?usp=drivesdk
If you take the file as it is and simply put in a bevel weight of 0.04 for ALL edges (you currently don't have one at the diagonal ones), it looks ok-ish.
Other than that, the topology simply isn't supporting what you are trying to do, those long polygons are bad. You could break them up by introducing a bunch of loop cuts all around the edges. But you also need to do something about that diagonal edge, like beveling it as well.
If you really want a super smooth object that looks nice even in an extreme close up, you might even have to go for subdiv modeling here.
Thanks for your help anyway, I'll try to do something with it.
In order to get a smooth look over faces, some interpolation of normals has to happen in the background. That works well for more regular meshes and not as well for things like e.g. extremely stretched polygons or sudden changes, as you have here with those very thin, long polygons and the sharp edge in the diagonal.
Feel free to look into the maths of normal interpolation if you are interested. It's complex, though.
So for a modeler, it's mostly a matter of practice to get a feeling for what works (or what needs to be done if it doesn't).