What word/phrase can I use to describe the activity of hiding a stupid mistake/incident, such as quickly cleaning up tea spilled by accident before anyone could notice?
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Not a single word, but one phrase used for this situation is to sweep it under the rug.
This is called a:
The definition in Merriam-Webster talks about illegal and unethical acts but this can be more widely used to "cover up" anything from small to large.
Note that "cover up" can also be used as a verb which I think is what OP is looking for.
You might use furtive, an adjective, or furtively, an adverb, to describe to wiping up.
This carries the sense of being done secretly or by stealth.
I don't think there is a single word or common phrase for what you are describing that doesn't also carry additional meaning that you may not want.
The other solutions offered so far (cover-up, sweep under the rug) carry the added implication of impropriety, and in fact, neither one necessarily means the problem has even been truly solved, only hidden.
The closest words that I would recommend would be those that mean to fix the problem that carry the added connotation of putting things back to the previous state, making a perfect repair.
These would include (depending on the kind of problem):
I would couple one of these words with something that indicates that it's happened before anyone else can see it.
In a more corporate/organizational context, you may want to use something like "damage control" or "disaster management," but that doesn't necessarily imply fixing the problem before it does any damage, more about just minimizing the damage and fixing things as well as possible as fast as possible.
Finally, for something like spilled tea, you could use descriptions of the action of fixing it to suggest the kind of immediacy you are looking for. If this is a narrative, you could say that he used his shirt to wipe it up and then looked around to see if anyone noticed, as an example. Whatever the particular problem, there are going to be options like this.