That is actually not clover, but wood sorrel, aka sourgrass. Fun fact: you can eat the leaves. As you might guess from the common name, they taste sour. https://www.wildedible.com/wild-food-guide/wood-sorrel
To be quite honest I'm not an expert, I think I've just struck lucky. I bought a tiny plant from B&Q (UK DIY shop) in December and put it in my kitchen windowsill. It gets sun for most of the day and I water it with tap water (:o) whenever it feels dry. I've recently repotted to a slightly bigger pot and the traps have really taken off again. Other than that, I have no advice! But I hope it helps.
You could maybe support it, but I would take the piece off and plant it.
I'd rather take the piece off as well; my only concern was not to make the plant suffer. Will it be able to keep on growing from the cut top?
Sure. One of the branches might take over as the main branch. You might have several main branches as well. If you want one, you could also cut the rest off. I would just leave it alone after you cut the top off.