No, we shouldn't cut cat's moustache.
''With regard to cats eating food or drinking water, that does not make the food or water naajis (impure), because of the report narrated in Sunan Abi Dawood and elsewhere, which says that a woman brought some hareesah (a kind of food) to ‘Aa’ishah (may Allaah be pleased with her) and found her praying. (‘Aa’ishah) gestured to her that she should put it down. Then a cat came and ate some of it. When ‘Aa’ishah had finished praying, she ate from where the cat had eaten and said, “The Messenger of Allaah (peace and blessings of Allaah be upon him) said: ‘They (cats) are not naajis (impure), rather they are among those who go around among you (al-tawwaafeena ‘alaykum).’ I saw the Messenger of Allaah (peace and blessings of Allaah be upon him) doing wudoo’ with water from which a cat had drunk.”
According to another report narrated by Abu Dawood (68) from Kabshah bint Ka’b ibn Maalik, who was married to Ibn Abi Qutaadah, Abu Qutaadah entered and she poured some water for him to do wudoo’. A cat came and drank from the water, and he tipped the vessel for it so that it could drink. Kabshah said: “He saw me looking at him and he said, ‘Do you find it strange, O daughter of my brother?’ I said, ‘Yes.’ He said, ‘The Messenger of Allaah (peace and blessings of Allaah be upon him) said, “They (cats) are not naajis (impure), rather they are among those who go around among you (al-tawwaafeena ‘alaykum).”
These two reports were classed as saheeh by al-Bukhaari, al-Daaraqutni and others, as was stated in al-Talkhees by Ibn Hajar, 1/15.
“Those who go around among you (al-tawwaafeena ‘alaykum)” means that they are like servants who are with people in their houses, i.e., cats are always with people, in their houses and among their vessels and furnishings, etc., and this is something that cannot be helped.
So if a cat drinks from a vessel or eats some of the food, it does not become naajis (impure). The owner has the choice. If he is not put off or if he needs it, he may eat (the food) or drink (the water), because it is taahir (pure), unless it is obvious that it will cause harm.
If he is put off by the idea of eating or drinking, then he may leave it.''
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In the name of Allah, the most Compassionate and Merciful
Cat saliva is not considered filth because of the hadith that Abu Qatada (Allah be pleased with him) related, that the Messenger of Allah (Peace be upon him) said (about the cat):
It is not impure, it is of those [creatures] that mingle with you. [Reported by Tirmidhi, Abu Dawud, Nasai, and Ibn Maja]
It is mentioned in Hashiyat al-Tahtawi:
If a cat drinks from a container, however, or licks ones hand before praying, then it would be mildly disliked (makruh tanzihan) to drink from that same container, or to pray without washing off the traces of saliva from ones hand. This is if one suspects it to have eaten filth or is uncertain of its statewhether impure (from eating filth) or pure. If a cat eats a mouse and drinks water immediately after, then the water becomes impure. However, by licking its mouth for a short while, the mouth is purified and it is no longer impure.
[Tahtawi 1:18, Dar al-Iman]
And Allah alone gives success.