Impact of prayer for forgiveness (istighfar) on receiving a response to supplication (du‘a’)
Question

Is constantly praying for forgiveness a means of receiving a response from Allah, may He be exalted, to a person’s supplication?
Praise be to Allah.Firstly:
By His kindness and mercy, Allah, may He be exalted, answers people’s supplications, as He, may He be glorified, says (interpretation of the meaning):
{And your Lord says, “Call upon Me; I will respond to you.” Indeed, those who disdain My worship will enter Hell [rendered] contemptible} [Ghafir 40:60]
{And [mention] Noah, when he called [to Allah] before [that time], so We responded to him and saved him and his family from the great flood} [Al-Anbiya’ 21:76]
{And [mention] Job, when he called to his Lord, “Indeed, adversity has touched me, and you are the Most Merciful of the merciful.”
So We responded to him and removed what afflicted him of adversity. And We gave him [back] his family and the like thereof with them as mercy from Us and a reminder for the worshippers [of Allah]} [Al-Anbiya’ 21:83-84].
It was narrated from ‘Ubadah ibn as-Samit that the Messenger of Allah (blessings and peace of Allah be upon him) said: “There is no Muslim on earth who calls upon Allah in supplication but Allah will grant it to him, or avert an equivalent evil from him, so long as his supplication does not involve sin or severing ties of kinship.” A man among the people said: Then I will offer a great deal of supplication. He said: “Allah will grant more.” Narrated by at-Tirmidhi (3573); he said: This hadith is hasan sahih, and gharib with this isnad.
It was narrated from Abu Sa‘id that the Prophet (blessings and peace of Allah be upon him) said: “There is no Muslim who offers supplication in which there is no sin or severing ties of kinship, but Allah will grant him one of three things in response to it: either He will hasten for him what he asked for, or He will store it for him in the hereafter, or He will avert an equivalent evil from him.”
They said: Then we will offer a great deal of supplication. He said: “Allah will grant more.” Narrated by Imam Ahmad in al-Musnad (17/213). Shaykh al-Albani said: It is hasan sahih, as noted in Sahih at-Targhib wa’t-Tarhib (2/278).
Secondly:
There are places and times in which a person is more likely to receive a response to his supplication. We have listed some of them in the answer to question no. 22438. They include:

Supplication on Laylat al-Qadr.
Supplication in the last part of the night.
Supplication when in prostration.
Supplication between the adhan and iqamah.
Supplication at the time of the adhan for the prescribed prayers, and when the Muslim army engages the enemy in battle.
Supplication of one who is oppressed.
Supplication after the sun has passed the meridian, before Zuhr.

Thirdly:
Prayer for forgiveness (istighfar) brings great benefits which it is good for the Muslim to learn about. In the Book of Allah, may He be exalted, Hud said to his people (interpretation of the meaning):
{And O my people, ask forgiveness of your Lord and then repent to Him. He will send [rain from] the sky upon you in showers and increase you in strength [added] to your strength. And do not turn away, persisting in wickedness.
O my people, seek the forgiveness of your Lord and repent to Him; He will send down upon you abundant rain from the sky, and will add strength to your strength. So do not turn away in sin} [Hud 11:52].
And before him, Nuh said (interpretation of the meaning):
{He will send [rain from] the sky upon you in [continuing] showers,
And give you increase in wealth and children and provide for you gardens and provide for you rivers} [Nuh 71:11-12].
We have mentioned in the answers to questions no. 39775 and 104919 the virtue and benefits of praying for forgiveness.
Praying for forgiveness makes it more likely that the believer will receive a response to his supplication in several ways:
1.. Praying for forgiveness erases sins, which makes a person closer to Allah, may He be exalted, and being close to Allah is one of the means of receiving a response to supplication.
Ibn al-Qayyim said: I said to Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) one day: One of the scholars was asked: Which is more beneficial to a person, glorifying Allah (tasbih) or praying for forgiveness (istighfar)? He said: If the garment is clean, then bukhur (incense) and rosewater are better for it, and if it is dirty, then soap and hot water are better for it. Then he (may Allah have mercy on him) said to me: What about if the garment is still dirty?"(Al-Wabil as-Sayyib  1/233).
2.. Praying for forgiveness is one of the means of increasing provision, and supplication and the response to it are provision from Allah, may He be glorified and praised.
Ibn Taymiyah said: I would have an issue or a problem that I found difficult to find an answer to, so I would ask Allah, may He be exalted, for forgiveness one thousand times, or more or less, until I felt at ease and found an answer to the issue or problem that was on my mind."(Al-‘Uqud ad-Duriyyah fi Manaqib Ibn Taymiyah  p. 10).
Fourthly:
What supports the idea that praying a great deal for forgiveness is one of the means of receiving a response to one’s supplication is the words of the Prophet (blessings and peace of Allah be upon him):
“The supplication of Dhun-Noon which he offered when he was in the belly of the fish: ‘La ilaha illa anta, subhanaka, inni kuntu min az-zalimin (There is no god worthy of worship except You; exalted are You. I have indeed done wrong’ [al-Anbiya’ 21:87]. No Muslim calls upon his Lord with it for anything but he will receive a response.”
Narrated by Ahmad (1462) and al-Tirmidhi (3505).
Al-Qurtubi said, commenting on the passage,
{And [mention] the man of the fish, when he went off in anger and thought that We would not decree [anything] upon him. And he called out within the darknesses, “There is no god worthy of worship except You; exalted are You. I have indeed done wrong.”
So We responded to him and saved him from the distress. And thus do We save the believers} [Al-Anbiya’ 21:87-88]:
In this verse, Allah promises the one who calls upon Him in supplication that He will respond, as He responded to Dhu’n-Nun and that He will save him as He saved Dhu’n-Nun; that promise is { And thus do We save the believers }."(Al-Jami‘ li Ahkam al-Qur’an  11/334).
That is because his supplication combines all types of Tawhid:

Affirmation of the oneness of Allah, may He be blessed and exalted.
Declaring Him to be far above any injustice, because His justice and wisdom are perfect.
Acknowledging his sin.
Observing proper etiquette when asking, as he [Dhu’n-Nun] did not ask of Allah, may He be exalted, by means of a clear statement; rather he only hinted at his situation. End quote.

See: Sharh ad-Du‘a’ min al-Kitab wa’s-Sunnah (p. 81).
For more information, please see the answer to question no. 318430 .
This idea is also confirmed by what al-Qurtubi (may Allah have mercy on him) mentioned when he said: This verse and the verse in Surat Hud indicate that prayer for forgiveness is a means by which provision and rain may come down.
Ash-Sha‘bi said: ‘Umar went out to pray for rain, and he did no more than pray for forgiveness before he came back, then it rained. They said: We did not see you pray for rain. He said: I have indeed prayed for rain by words which are the stirrers-up of rain. Then he recited the words (interpretation of the meaning):
{ And said, ‘Ask forgiveness of your Lord. Indeed, He is ever a Perpetual Forgiver.
He will send [rain from] the sky upon you in [continuing] showers} [Nuh 71:10-11].
Al-Awza‘i said: The people went out to pray for rain, and Bilal ibn Sa‘d stood up to address them. He praised and glorified Allah, then he said: O Allah, we have heard Your words (interpretation of the meaning) {There is not upon the doers of good any cause [for blame]} [at-Tawbah 9:91]. We admit our wrongdoing, and Your forgiveness is not for anyone except people like us. O Allah, forgive us, have mercy on us, and grant us rain. He raised his hands and the people raised their hands, then it rained.
Ibn Subayh said: A man complained to al-Hasan of drought, and he said to him: Ask Allah for forgiveness. Another man complained to him of poverty, and he said to him: Ask Allah for forgiveness. Another man said to him: Pray to Allah to bless me with a child. He said to him: Ask Allah for forgiveness. Another man complained to him that his garden was dry, and he said to him: Ask Allah for forgiveness. We asked him about that, and he said: I was not speaking of my own accord. Allah, may He be exalted, says in Surat Nuh (interpretation of the meaning):
{Ask forgiveness of your Lord. Indeed, He is ever a Perpetual Forgiver.
He will send [rain from] the sky upon you in [continuing] showers
And give you increase in wealth and children and provide for you gardens and provide for you rivers} [Nuh 71:10-12].
And we have already seen in Surat Al ‘Imran how to seek forgiveness, and that that should be done with sincerity and giving up sins. This is the most important means of receiving a response."(Tafsir al-Qurtubi  18/302-303).
And Allah knows best.