Is it sufficient to cleanse impurities from the face by wiping them with a wet hand without pouring water, and what is the ruling on prayers performed in that state?

It is not sufficient to put water by hand and wipe the spot of impurity to purify it. Rather, water must be poured over the impurity until it covers it and removes its trace, whether color or odor. What the questioner was doing is not sufficient, but his previous prayers are valid, and he does not have to repeat them if he prayed forgetfully or out of ignorance.