Dua When One Sees Someone Sneezing

Saying “Yarhamuk Allah (may Allah grant you mercy)” to the one who sneezes, praises Allah (saying “al-hamdu Lillah” after sneezing) is part of the etiquette prescribed in Islam and the rights that one Muslim has over another.
See answers to questions no. 67805 and 178639
But if he does not say “al-hamdu Lillah”, he does not have the right to ask another person to tell him “Yarhamuk Allah”.
Al-Bukhaari (6221) and Muslim (2991) reported that Anas (may Allah be pleased with him) said: Two men sneered before the Prophet (peace and blessings of Allaah be upon him), and he told Yarhamuk Allah (may Allah have mercy on you) for one of them and not for the other. Something was said to him about it and he said: “This one praised Allah (said Al-hamdu Lillaah), but this one did not praise Allah.” Abu Burdah reportedly said: I entered Abu Moosa when he was in the house of the daughter of al-Fadl ibn ‘Abbaas and I sneezed, but he did not tell me Yarhamuk Allah, but she sneezed and he said it. to her.
I went back to my mother’s house and told her. When he came to her home, she said: My son sneezed in your presence and you did not say Yarhamuk Allah, but she sneezed and you told her so. He said: Your son sneezed but he did not praise Allah. I did not tell him Yarhamuk Allah. She sneezed and she praised Allah, so I told him Yarhamuk Allah. I heard the Messenger of Allah (peace and blessings of Allaah be upon him):
“When one of you sneezes and praises Allah, tell him Yarhamuk Allah, but if he does not praise him, do not tell him. . ” Reported by Musulman (2992)
An-Nawawi (may Allaah have mercy on him) says: This clearly indicates that it is enjoined to say Yarhamuk Allah if the one who sneezes praises Allah, and it is not permitted to tell him Yarhamuk Allah if he does not praise Allah. that’s why makrooh told him if he did not praise Allah. But if he praises Allah, but the other person did not hear him, then he does not have to tell him Yarhamuk Allah. Maalik says: He should not tell him Yarhamuk Allah if he does not hear him praise Allah. But if you see that someone next to him says Yarhamuk Allah, you should also say it.
End of quote by Sharh an-Nawawi “ala Muslim” (18/121)
See also answer to question no. 3448 and 106432.

https://islamsciences.com/duas/dua-after-salah-tasbeeh/

Secondly:

If the child has reached the age of discernment and is able to understand if you teach him good manners, then it is recommended to teach him to praise Allah if he sneezes, because it is proven that the Prophet (blessings and salvation) be on him)) taught the boy the etiquette to eat and asked the Muslims to teach their children how to pray, etc.
Children and those new to Islam should be taught to say “al-hamdu Lillah”. Likewise, those who grew up in the far-off desert should be taught, because those who grew up in such circumstances are expected to be unaware of it. If a child sneezes and praises Allah, we can say,
“May you be blessed” or “May Allah help you” or Yarhamuk Allah. “This has been declared by Sheikh Abd al-Qaadir. End of the quote by Mataalib Ooli an-Nuha (1/945).
See also Kashshaaf al-Qinaa ‘(2/158) What seems to be the case is that the right of one who praises Allah (after sneezing) applies to young and old, without any distinction, and in both cases should be said the same sentence:
“Yarhamuk Allah”.
It is said in Fath al-Mu’een (4/219) In Al-Adhkaar, an-Nawawi did not tell the difference between young and old with regards to saying Yarhamuk Allah (to whoever praises Allah after sneezing).
End of the quote by Fath al-Mu’een Sharh Alfaaz al-Mu’een (4/219)

Thirdly:

If the child is small and has not yet reached the age when he can learn and can not yet pronounce the words “al-hamdu Lillah” correctly, some scholars are of the opinion that his guardian should praise Allah in his name. but what is more likely is that it is only required of one who sneezes and that it should not be done by someone else in his name. But if a person offers a supplication (du’aa) to the child by saying all that is appropriate, such as asking for mercy for him or becoming just, etc., then we hope there is no There is nothing wrong with that, because the child is not negligent in failing to praise Allah, and du’aa ‘for such is generally prescribed. But this should not be taken as a regular practice.
Ibn Muflih (may Allah be merciful to him) told al-Adaab ash-Shar’iyyah (2/343): Abdullah ibn Ahmad told al-Hasan that he had been interrogated about a small child who sneezes.
He said: We must tell him: May you be blessed. The author of an-Nazm said: If a child sneezes, he may be asked to say al-hamdu Lillah.
Then it should be said: Yarhamuk Allah or May you be blessed, etc., and we should teach him to answer. If he is small, his guardian or other present person may praise Allah, and phrases similar to those mentioned above can be told to the child.
Ibn Muflih said: With regard to the teaching of the response to the child, this is similar to the question of returning the welcome message (ie both elements – the welcome message and the answer – do not have to be combined). But what seems to be the case, according to the opinion of other scholars, is that duaaa can be offered to the child, even if he does not say Al-hamdu Lillah. But it can be said that dunaa is akin to saying Yarhamuk Allah, so it should be connected to him saying Al-hamdu Lillah, as in the case of an adult.
But the first point of view is more likely. And Allah knows better. End quote from Al-Adaab ash-Shar’iyyah (2/327)

Fourthly

If a person sneezes while praying, he is enjoined to praise Allah, but he should not disturb others by doing so or distract them from their prayers. If he sneezes and does not praise Allah, it does not matter.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said:
If a person sneezes, whether during prayer or otherwise, it is not mandatory to say Al-hamdu Lillah, but it is preferable and more appropriate. But if he does not praise Allah, he does not sin.
Praying Allah after sneezing is prescribed to those who pray and do not pray, unless he prays and is afraid to disturb other worshipers. In this case, he should praise Allah gently and not say it aloud. is said aloud, it will disturb the other faithful or bring someone to say Yarhamuk Allah (reflex). If someone Say to him who sneezes and then praises Allah, “Yarhamuk Allah”, and to him who says that he is praying at that time, his prayer is invalidated. End of the quote by Fataawa Noor ‘ala ad-Darb (5/313) See also answer to question no. 106435
And Allah knows better.