To clean impurity, is it sufficient to wipe it several times with a damp cloth?

If a child urinates on the carpet and the like, in order to remove the impurity it is sufficient to use a sponge or cloth to mop up the urine, then wash it by putting water on it, and you should repeat that until you think it likely that the impurity has been removed. That will make the matter of pouring water on the carpet easier, because what is poured onto it will be a small amount, on the site of the impurity only.

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: How can one clean a large carpet of impurity? Is squeezing when washing away impurity required after removing the substance itself?

He replied: The way to wash a large carpet and clean it of impurity is to remove the substance of the impurity first, if it has any substance. If it is solid, then it can be picked up. If it is liquid, like urine, it may be dried up with a sponge until it has all been absorbed, then after that water should be poured on it until you think that the traces of it have been removed, or that the impurity has been removed. In the case of urine, this may be achieved by doing that two or three times. As for squeezing, it is not essential, unless the impurity cannot be removed except in this manner, such as if the impurity penetrated deeply into the thing that is being washed, and it is not possible to clean the inside of it except by squeezing. In that case, it must be squeezed. End quote from Fataawa Noor ‘ala ad-Darb.

But if that is on a hard surface such as tile, then the matter is easier, because hard surfaces do not absorb impurity, so if you clean the place with a damp towel or cloth, then wash it a few times, that will be sufficient to purify it, on condition that all traces of the impurity, such as colour and smell, disappear.

And Allah knows best.