Yeah, just that!

Shalom, you may clear the whole box into the garbage, and spill new litter in. The reason is that the litter is ready to be used and is not considered muktze since you dedicated it for this use (see Shulchan aruch 308;38, shemirat shabbat kehilchata 20;46, comment 175). And after the cat dirties it, it is permissibale to clear it, because it has a bad smell and is considered a "graf shel rei" meaning that one is allowed to clear muktze if it is has a very unpleasant sight or smell (Shulchan aruch 308;34). However just taking out the dirty part and leaving the rest is considered "borer"- separating the dirty part from the clean litter. To solve this problem, you can either clear the whole box or take out with the dirty part also some of the litter, and it that case you wouldn't be doing borer (beur halcacha 319;4). All the best!