Is it necessary to renew wudu in order to offer the prayer of repentance?
Question

Is renewing wudu one of the conditions of the prayer of repentance even if I am already in a state of purity?
Praise be to Allah.Firstly:
Islam encourages the one who repents to do wudu, pray two rak`ahs and seek forgiveness.
It was narrated that `Ali (may Allah be pleased with him) said: I was a man who, if I heard a hadith from the Messenger of Allah (blessings and peace of Allah be upon him), Allah would cause me to benefit from it as much as He willed, but if one of his companions told me about it, I would ask him to confirm it with an oath, then if he swore an oath I would believe him.
Abu Bakr told me – and Abu Bakr spoke the truth – saying: I heard the Messenger of Allah (blessings and peace of Allah be upon him) say: “There is no one who commits a sin, then purifies himself well, then stands up and prays two rak`ahs, then asks Allah for forgiveness, but Allah will forgive him.” Then he recited this verse (interpretation of the meaning):
{And those who, when they commit an immorality or wrong themselves [by transgression], remember Allah and seek forgiveness for their sins – and who can forgive sins except Allah? – and [who] do not knowingly persist in what they have done}
[Al `Imran 3:135].
Narrated by Abu Dawud, 152; and by at-Tirmidhi, 406, who said: The hadith of `Ali is a sound hadith. Shaykh al-Albani classed its isnad as authentic in Sahih Sunan Abi Dawud, 5/252.
That is aimed at achieving perfect repentance and removing the traces of sin, because each of these actions, according to Islamic teachings, is a means of erasing sin, and each of them has been enjoined in specific terms.
Regarding wudu, it is narrated in the hadith of Abu Hurayrah that the Messenger of Allah (blessings and peace of Allah be upon him) said: “When a Muslim – or a believer – does wudu and washes his face, every sin that he looked at with his eyes comes out from his face with the water – or with the last drop of the water. When he washes his hands, every sin that he committed with his hands comes out from his hands with the water – or with the last drop of the water. When he washes his feet, every sin to which he walked with his feet comes out from his feet with the water – or with the last drop of the water – until he emerges cleansed of sin.” Narrated by Muslim, 244.
It was narrated that `Uthman ibn `Affan said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever does wudu and does wudu well, his sins come out of his body, even from beneath his nails.” Narrated by Muslim, 245.
Regarding prayer, Allah, may He be exalted, says (interpretation of the meaning):
{And establish prayer at the two ends of the day and at the approach of the night. Indeed, good deeds do away with misdeeds. That is a reminder for those who remember}
[Hud 11:114].
Regarding seeking forgiveness, Allah, may He be exalted, says (interpretation of the meaning):
{And those who, when they commit an immorality or wrong themselves [by transgression], remember Allah and seek forgiveness for their sins – and who can forgive sins except Allah? – and [who] do not knowingly persist in what they have done.
Those – their reward is forgiveness from their Lord and gardens beneath which rivers flow [in Paradise], wherein they will abide eternally; and excellent is the reward of those who work [and strive]!}
[Al `Imran 3:135-136].
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
The texts of the Qur’an and Sunnah indicate that the punishment of sin may be waived by approximately ten means, the first of which is repenting; there is consensus regarding that among the Muslims…
The second means is seeking forgiveness, as it was narrated in al-Bukhari and Muslim from the Prophet (blessings and peace of Allah be upon him) that he said: “If a person commits a sin, then says: ‘O Lord, I have committed a sin; forgive me,’ Allah says: ‘My slave knows that he has a Lord Who forgives sins and brings people to account for them; I have forgiven My slave’…”
In Sahih Muslim it is narrated that the Prophet (blessings and peace of Allah be upon him) said: “If you did not repent, Allah would do away with you and would bring people who would commit sin then ask for forgiveness, and He would forgive them.”
End quote from Majmu` al-Fatawa, 7/487-488.
Generally speaking, if one does good deeds after bad deeds, there is the hope that they will be erased. It was narrated that Abu Dharr said: The Messenger of Allah (blessings and peace of Allah be upon him) said to me: “Fear Allah wherever you are; follow a bad deed with a good deed, and it will erase it; and have a good attitude towards people.” Narrated by at-Tirmidhi, 1987; he said: This is a good, authentic hadith.
Secondly:
If a sinner has wudu at the time when he commits the sin, and nothing comes out of him that would invalidate his wudu, then in this case it is not stipulated that this sinner should repeat his wudu in order to pray two rak`ahs, but it is recommended for him to do that. Committing sin does not invalidate wudu.
Ibn al-Mundhir (may Allah have mercy on him) said:
When a person purifies himself, he remains in a state of purity unless there is evidence to indicate that his state of purity has been invalidated.
All of the scholars of the regions from whom we acquired knowledge are unanimously agreed that slander, lying and backbiting do not invalidate a state of purity or make it obligatory to do wudu.
This is also the view of the scholars of Madinah and of Kufa, ahl ar-ra’y and others. And it is the view of ash-Shafa`i, Ahmad and Is-haq.
It was narrated from Ibn `Abbas that he was asked: do stealing, treachery, lying, immoral actions and looking at that which is not permissible make it obligatory to do wudu? He said: No; there are two types of impurity, that which results from sins of the tongue [and does not require wudu] and that which results from bodily functions [which does require wudu]…
It was narrated from more than one of the earlier scholars that they used to enjoin wudu for one who uttered foul speech and offended the Muslims…
But I do not think that those who enjoined wudu enjoined it except by way of recommendation; that is clear from the wording of their statements.
End quote from al-Awsat, 1/230-233.
For more information on things that invalidate wudu, please see the answer to question no. 14321.
Based on that, if this person prayed without renewing his wudu, his prayer is valid.
And Allah knows best.