Praise be to Allah.
In the answer to question no. 158204 we stated that if a person makes a mistake in reciting a soorah, or forgets something of it, whilst he is praying, then he should try to correct the mistake and remember what he forgot. If he is not able to do that, then he may skip that verse and move on to the next one, or he may stop reciting that soorah and start reciting a different one.
But if he adds to the Qur’an something that is not part of it deliberately – whether that is during the prayer or otherwise, then this is emphatically forbidden and the scholars have stated that whoever omits a letter of the Qur’an or substitutes a different letter or adds a letter has committed an act of kufr.
Al-Qaadi ‘Iyaad (may Allah have mercy on him) said:
The Muslims are unanimously that the Qur’an which is recited in all corners of the earth, and is written in the Mushaf between the two covers by the hands of the Muslims, from the beginning of Soorat al-Faatihah to the end of Soorat an-Naas is indeed the word of Allah and His Revelation that He sent down to His Prophet Muhammad (blessings and peace of Allah be upon him); that everything in it is truth; and that anyone who deliberately omits a letter of it, or substitutes one letter for another, or adds to it a letter that is not included in the Mushaf on which there is consensus, and there is consensus that it is not part of the Qur’an, has committed an act of kufr.
End quote from ash-Shifa, 2/304-305; see also at-Taqreer wa’t-Tahbeer by Ibn Ameer al-Haaj, 2/215
It says in al-Mawsoo‘ah al-Fiqhiyyah, 35/214: The Qur’an is the word of Allah, the miracle that was sent down to the Messenger of Allah (blessings and peace of Allah be upon him), and was transmitted by means of tawaatur (i.e., it was narrated from so many people by so many people that it is inconceivable that they could all have agreed upon a lie). It is haram to deliberately make mistakes in it, whether that changes the meaning or not, because its words and phrases are tawqeefi (i.e., to be accepted and recited as they were revealed, and they cannot be altered) and it was transmitted to us by means of tawaatur. It is not permissible to change any word of it by altering the vowels or changing the letters, replacing one letter with another. End quote.
Based on that, it is not permissible for a student to write a word or a letter that he knows is not part of the Qur’an or that does not belong in this verse; rather he should either try hard to remember, or he should leave the parts that he has forgotten blank. He can also apologize for that by explaining that he forgot where this word goes and that he did not want to write something of which he was not certain in this case.
And Allah knows best.