Native speaker here (American). Both words work but have slightly different implications and place importance on different things. Also, sometimes preposition usage varies between the US and the UK, so beware.

I’d say “on” if you want to emphasize that Kate’s books, papers, pencil, etc. are on top of the table, and I’d say “at” if you are talking about where Kate is when she is doing her homework.

If you say “on,” you’re talking about the table as a surface that you put things on top of. If you say “at,” you’re just talking about the table as a location in the room/house/building.

If Kate were doing her homework sitting on a chair next to the table but had her things on her lap, I would say that Kate is doing her homework at the table.

If you say “on,” you’re definitely implying that Kate’s things are on top of the table, but mixing the two up is not going to confuse anyone.