Kindly refer to following video
https://youtu.be/VNtvCouwAlg

While writing hadith give reference.
Here is response from Mushafiq Sultan sb for similar question

According to Ustaz Ghamidi’s understanding, a hadith report by its very nature cannot become an independent source for deriving any general ruling unless it has its basis in the two primary sources, the Qur’an and the Sunnah. The reason for this is the obvious nature of the transmission of this historical record, which the Holy Prophet (sws) never made any effort to preserve and disseminate. Also, it is possible that while narrating an episode, some important parts related to it are not transmitted, which could otherwise change its implication.

People cite the following Aayah of the Qur’an to argue that Muslims are forbidden from seeking forgiveness for non-Muslims.

مَا كَانَ لِلنَّبِيِّ وَالَّذِينَ آمَنُوا أَنْ يَسْتَغْفِرُوا لِلْمُشْرِكِينَ وَلَوْ كَانُوا أُوْلِي قُرْبَى مِنْ بَعْدِ مَا تَبَيَّنَ لَهُمْ أَنَّهُمْ أَصْحَابُ الْجَحِيمِ

It is not proper for the Prophet and those who believe to ask God’s Forgiveness for the mushrikun, even though they be of kin after it has become clear to them that they are the dwellers of the Fire. (9:113)

The concluding part of this Aayah is itself clear about the reason for asking the Prophet (sws) and the believers not to seek God’s forgiveness for the Mushrikun. Had the Aayah ended at ‘walau kaanu oolii qurbaa’, it could be considered an absolute prohibition. However, it adds the condition ‘after it has become clear to them that they are the dwellers of the Fire.’ This directive relates to the phase in which people refuse to accept faith even after God’s messenger has conclusively communicated the truth to them and announced his acquittal (Bara’ah). The supplications of God’s messenger and his companions are a guarantee for the refuge and security of the nation. In this phase of reward and punishment, it was essential that they be divested of this refuge so that they are ready to face the punishment that had become destined for them. However, after the passing away of Muhammad (sws) there is no possibility of knowing whether such conclusive communication of the truth has happened for specific individuals or not. Thus, there is nothing wrong if the following words of Jesus (pbuh) are used to pray for the forgiveness of a deceased non-Muslim:

إِن تُعَذِّبْهُمْ فَإِنَّهُمْ عِبَادُكَ وَإِن تَغْفِرْ لَهُمْ فَإِنَّكَ أَنتَ الْعَزِيزُ الْحَكِيمُ

If You punish them, they are your servants, and if You forgive them, You only are Mighty and very Wise, (5:118).

Note that Jesus (pbuh) will use these words after coming to know that his followers took him and his mother as deities besides God.

Concerning the hadith report that you have asked about, there could be many possible reasons for not allowing the Prophet (sws) to seek forgiveness for his mother, which is not mentioned in this report. Also, the following hadith report is worthy of consideration:

Narrated Ibn ‘Abbas:

“I heard ‘Umar bin Al-Khattab saying: ‘When ‘Abdullah bin Ubayy died, the Messenger of Allah (ﷺ) was called to perform the funeral prayer over him. The Messenger of Allah (ﷺ) came to him, and when he stood over him, about to perform the prayer, he turned until he was standing at his chest. I said: “O Messenger of Allah! (You pray) for Allah’s enemy ‘Abdullah bin Ubayy, who on this day said this and that” – mentioning different days. He said: “The Messenger of Allah (ﷺ) was smiling until I had said too much to him and he said: ‘Leave me O ‘Umar! Indeed I have been given the choice, so I chose. I was told: Whether you ask forgiveness for them, or do not ask for forgiveness for them. Even though you ask for their forgiveness seventy times, Allah will not forgive them (9:80). If I knew that were I to ask more than seventy times that he would be forgiven, then I would do so.” He said: “Then he performed the Salat for him and walked with him (his funeral procession) and he stood at his grave until it was finished. I was amazed at myself and my daring to talk like that to the Messenger of Allah (ﷺ), while Allah and His Messenger (ﷺ) know better. But by Allah! It was not long until these two Ayat were revealed: ‘And never pray for any of them who dies nor stand at his grave… (9:84) until the end of the Ayah. He said: “So afterwards the Messenger of Allah (ﷺ) did not perform the Salat for a hypocrite, nor would he stand at his grave until Allah took him.”

(Tirmidhi, Kitab al-Tafsir, Hadith 3097)

In 9:84, the Holy Prophet (sws) was forbidden from offering the funeral prayer of the hypocrites and standing on their graves. Obviously, there is an indication here that when the Holy Prophet (sws) was asked to avoid the graves of the hypocrites, how could he visit the graves of any polytheist (in this case, his mother supposedly) if they were the denizens of hell?

Even if we agree that the mother of the Holy Prophet (sws) was a denizen of hell and therefore he was forbidden from seeking forgiveness for her, that still does not make it a general rule, which can be extrapolated to all the non-Muslims.

From the narrators’ point of view:

All the versions of this Hadith that are narrated in the more reliable books of Hadith like Muslim, Sunan Abi Dawud and Sunan Nasayee include Yazid ibn Kaysaan who (as written in the chains of narrators of this Hadith) has narrated it from Abi Hazim who has narrated it from Abu Huraira.

Although many scholars of Rijal have approved Yazid ibn Kaysaan, there are in fact some serious comments about him in the books of Rijal that make his position as a narrator of Hadith (especially when he is the only person who has narrated this from Abu Huraira) very problematic:

Al-Jurjani has included him in his book al-Kamil fi Zu’afa al-Rijal (Collection of weak narrators), entry 2180. He narrates from Yahya and Marwan that Yazid ibn Kaysaan is not one that could be trusted.

Ibn al-Jawzi has included him in his book al-Zu’afa wa al-Matrukin (Weak and abandoned), entry 3801.

Some of the above are also reported in the book al-Tahzeeb al-Kamaal by a-Mazi (no. 7041). Also, we read:

Abd al-Rahman ibn Abi Hatam reports that his father (Abi Hatam) considered the Ahadith of Yazid ibn Kaysaan to be worth of writing and did not consider him weak however when he asks Abi Hatam “(so) can we reason using his Hadith”, Abi Hatam replies: “No!”. He further writes that Bukhari has indeed added his name in his book of Zu’afa – i.e. weak narrators (although he also writes that his father was of the opinion that he should not have listed as weak).

Based on the above points, we have no reason to derive a ruling that one cannot pray for the forgiveness of a non-Muslim.

Link: http://www.al-mawrid.org/index.php/questions/view/seeking-forgiveness-for-non-muslims