There is no specific du‘aa’ for each day and night of Ramadan

Praise
be to Allah.

Firstly: 

Ibn Khuzaymah narrated in his Saheeh (1887) that Salmaan (may Allah
be pleased with him) said: The Messenger of Allah (blessings and peace of
Allah be upon him) addressed us on the last day of Sha‘baan and said: “O
people, a great month has come upon you, a blessed month… It is a month the
beginning of which is mercy, its middle is forgiveness and its end is ransom
from the Fire.” 

In
the answer to question no. 21364, we explained that
this hadith is da‘eef (weak).  

The entire month of Ramadan is mercy from Allah; the entire month is also
forgiveness and ransom from the Fire. None of these blessings is restricted
to any one part of the month to the exclusion of any other part, and this is
a reflection of the vastness of Allah’s mercy. 

Muslim (1079) narrated that Abu Hurayrah (may Allah be pleased with him)
said: The Messenger of Allah (blessings and peace of Allah be upon him)
said: “When Ramadan comes, the gates of Paradise are opened and the gates of
Hell are closed, and the devils are fettered.” 

At-Tirmidhi (682) narrated that Abu Hurayrah said: The Messenger of Allah
(blessings and peace of Allah be upon him) said: “When the first night of
Ramadan comes, the devils and mischievous jinn are chained up, and the gates
of Hell are closed, and none of its gates are opened. The gates of Paradise
are opened and none of its gates are closed. And a caller cries out: ‘O
seeker of good, proceed; O seeker of evil, desist.’ And Allah has people
whom He frees (from the Fire), and that happens every day.”

Classed as saheeh by al-Albaani in Saheeh at-Tirmidhi. 

Based on that, singling out the first third of Ramadan to pray for mercy,
the second third to pray for forgiveness and the last third to pray for
ransom from Hell is an innovation for which there is no basis in Islamic
teachings. There is also no justification for singling out these times for
these supplications, because all the days of Ramadan are equal in that
regard. Rather the Muslim may pray for whatever he wants of goodness in this
world and the hereafter throughout Ramadan; that includes asking Allah for
mercy, forgiveness, ransom from the Fire and admission to Paradise. 

Secondly: 

The Muslim should offer a great deal of du‘aa’, asking for goodness and
mercy, especially in this month, making the most of this time of goodness
and blessing (barakah), seeking the mercy and forgiveness of their Lord, may
He be exalted. Allah, may He be exalted, says (interpretation of the
meaning):

“And when My
slaves ask you (O Muhammad blessings and peace of Allah be upon him)
concerning Me, then (answer them), I am indeed near (to them by My
Knowledge). I respond to the invocations of the supplicant when he calls on
Me (without any mediator or intercessor). So let them obey Me and believe in
Me, so that they may be led aright”

[al-Baqarah 2:186]. 

Ibn Katheer (may Allah have mercy on him) said: 

The fact that Allah, may He be exalted, mentions this verse, which
encourages one to offer du‘aa’, in the midst of the verses that speak of the
rulings on fasting, indicates that one should strive hard in offering du‘aa’
at the end of the fast and, indeed, every time one breaks the fast.

End quote from Tafseer Ibn Katheer (1/509) 

It
is good for the one who offers du‘aa’ to follow proper etiquette and to
recite many of the du‘aa’s that have been narrated from the Prophet
(blessings and peace of Allah be upon him), and not to overstep the mark in
offering du‘aa’. He should observe the etiquette of du‘aa’ and recite often
the du‘aa’s that it is recommended to recite a great deal in Ramadan and
also at times other than Ramadan. These include the following: 

·     

“Rabbana aatina fi’l-dunya hasanah wa fi’l-aakhirah
hasanah wa qina ‘adhaab al-naar (Our Lord, give us that which is good in
this world and that which is good in the Hereafter, and protect us from the
torment of the Fire),” [al-Baqarah 2:201]

·     
“Rabbana hab
lana min azwaajina wa dhurriyyaatina qurrata a‘yunin waj‘alna lil-muttaqeena
imaama (Our Lord! Bestow
on us from our wives and our offspring who will be the comfort of our eyes,
and make us leaders for the Muttaqoon (the pious))”
[al-Furqaan 25:74].

·     
“Rabb ij‘alni
muqeema as-salaati wa min dhurriyyati Rabbana wa taqabbal du‘aa’. Rabbana
ighfir li wa li waalidayya wa lil-mu’mineena yawma yaqoom ul-hisaab (O
my Lord! Make me one who performs As-Salat (Iqamat-as-Salat), and (also)
from my offspring, our Lord! And accept my invocation. Our Lord! Forgive me
and my parents, and (all) the believers on the Day when the reckoning will
be established)” [Ibraaheem
14:40-41].

·     

Allaahumma innaka ‘afuwwun tuhibb al-‘afwa
fa’affu ‘anni
(O Allaah, You are forgiving and You love forgiveness, so forgive me).

·     

Allaahumma inni as’aluka min al-khayri
kullihi ‘aajilihi wa aajilihi, ma ‘alimtu minhu wa ma lam a‘lam, wa a‘oodhu
bika min ash-sharri kullihi ‘aajilihi wa aajilihi, ma ‘alimtu minhu wa ma
lam a‘lam. Allahumma inni as’aluka min khayri ma sa’alaka ‘abduka wa
nabiyyuka, wa a‘oodhu bika min sharri ma ‘aadha minhu ‘abduka wa nabiyyuka.
Allahumma inni as’aluka al-jannata wa ma qarraba ilayha min qawlin aw ‘amal,
wa a‘oodhu bika min an-naari wa ma qarraba ilayha min qawlin aw ‘amal, wa
as’aluka an taj‘ala kulla qada’in qadaytahu li khayran
(O Allah, verily I ask you for all that is good in this world and the
hereafter, what I know of it and what I do not know, and I seek refuge with
You from all that is evil in this world and the hereafter, what I know of it
and what I do not know. O Allah, verily I ask You for the good of that which
Your slave and Prophet asked of You, and I seek refuge with You from the
evil of that from which Your slave and Prophet sought refuge with You. O
Allah, verily I ask you for Paradise and that which will bring me nearer to
it of words and deeds, and I seek refuge with You from Hell and that which
would bring me nearer to it of words and deeds. I ask You to make everything
You decree for me good).

·     

Allahumma inni as’aluka al-‘aafiyata
fi’d-dunya wa’l-aakhirah. Allahumma inni as’aluka al-‘afwa wa’l-‘aafiyata fi
deeni wa dunyaaya wa ahli wa maali. Allahumma astur ‘awraati wa aamin
raw‘aati. Allahumma ihfazni min bayna yadayya wa min khalfi wa ‘an yameeni
wa ‘an shimaali, wa min fawqi, wa a‘oodhu bi ‘azamatika an ughtaala min
tahti
(O Allah, I ask You for pardon and well-being in my religious and worldly
affairs, my family and my wealth. O Allah, conceal my faults and protect me
from that which causes me to worry. O Allah, protect me from before me and
from behind me, from my right and from my left, and from above me, and I
seek refuge in Your greatness lest I be destroyed from beneath me).

Similarly, it is recommended to recite general du‘aa’s from the Qur’an and
Sunnah, any du‘aa’ is good in which the individual strives hard, beseeching
his Lord. None of that is limited to Ramadan only. 

It
is also mustahabb to say after breaking the fast:
“Dhahaba al-zama’
wa abtalat al-‘urooq wa thabata al-ajr in sha Allah (Thirst is gone, the
veins are moistened and the reward is certain if Allah wills).” 

Please see also the answer to question no. 14103 and
26879. 

One should strive hard in du‘aa’ in the last third of the night in
particular, every night. 

Please see the answer to question no. 140434 

During the last ten days, one should recite a great deal the words: “Allaahumma
innaka ‘afuwwan tuhibb ul-‘afwa fa’fu ‘anni
(O Allaah, You are forgiving and love forgiveness, so forgive me).” 

Please see the answer to question no. 36832. 

Please see the answer to question no. 36902 for more
information on the etiquette of du‘aa’. 

And Allah knows best.