All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
If a person’s mouth becomes impure because of drinking wine or vomiting, or anything else, then he is obliged to purify his mouth by washing it with water until he predominantly believes that the impurity is removed. It is not a condition to wash it three times; rather, one must wash until he predominantly thinks that the impurity is removed.
If he knows that the remains of food in his mouth (like meat between the teeth) was stained with impurity, then he must remove it in order to make sure that all impurities are removed. However, if he only doubts about that the impurity touched the remains of food, then his mouth is, in principle, pure until the contrary is proven with certainty.
Al-Khateeb Ash-Shirbeeni said in Mughni Al-Muhtaaj:
"If one washes his impure mouth, he should exaggerate in gurgling in order to wash everything that is apparent (i.e., his entire mouth back to his throat). Also, he should not swallow any food or drink before washing his mouth so that he would not eat impurity. Imaam An-Nawawi quoted this in his book Al-Majmoo’ from Shaykh Abu Muhammad Al-Juwayni, and he [An-Nawawi] confirmed the ruling.” [End of quote]
Allah knows best.