You pray two rakaat then make dua during or after the prayer... there's no specific Dua,
'Uthman ibn Hunayf,narrates.
that a blind man came to the Prophet (Allah bless him and grant him peace) and said, "I've been afflicted in my eyesight, so please pray to Allah for me." The Prophet (Allah bless him and grant him peace) said: "Go make ablution (wudu), perform two rak'as of prayer....The Prophet (Allah bless him and grant him peace) added, "And if there is some need, do the same."
Abullah ibn Abi Awfa (Allah be pleased with him) relates that the Messenger of Allah (Allah bless him and give him peace) said, “Whoever has a need with Allah, or with any human being, then let them perform ritual ablutions well and then pray two rakats. After that, let them praise Allah and send blessings on the Prophet (Allah bless him and give him peace). After this, let them say,
لا إِلَهَ إِلا اللَّهُ الْحَلِيمُ الْكَرِيمُ
سُبْحَانَ اللَّهِ رَبِّ الْعَرْشِ الْعَظِيمِ
الْحَمْدُ لِلَّهِ رَبِّ الْعَالَمِينَ
أَسْأَلُكَ مُوجِبَاتِ رَحْمَتِكَ وَعَزَائِمَ مَغْفِرَتِكَ وَالْغَنِيمَةَ مِنْ كُلِّ بِرٍّ وَالسَّلامَةَ مِنْ كُلِّ إِثْمٍ
لا تَدَعْ لِي ذَنْبًا إِلا غَفَرْتَهُ وَلا هَمًّا إِلا فَرَّجْتَهُ وَلا حَاجَةً هِيَ لَكَ رِضًا إِلا قَضَيْتَهَا يَا أَرْحَمَ الرَّاحِمِينَ
There there no god but Allah the Clement and Wise.
There is no god but Allah the High and Mighty.
Glory be to Allah, Lord of the Tremendous Throne.
All praise is to Allah, Lord of the worlds.
I ask you (O Allah) everything that leads to your mercy, and your tremendous forgiveness, enrichment in all good, and freedom from all sin.
Do not leave a sin of mine (O Allah), except that you forgive it, nor any concern except that you create for it an opening, nor any need in which there is your good pleasure except that you fulfill it, O Most Merciful!”
The prayer of need is very simple: It is essentially to raise one's need to Allah Most High, by performing ritual ablution (wudu), praying 2 rakats (or four), and then making whole-hearted dua to Allah. If one uses the abovementioned Prophetic supplication, or other similar supplications transmitted from the Beloved Messenger of Allah (Allah bless him and give him peace), it is best. [Ibrahim al-Halabi, Sharh Munyat al-Musalli; al-Fatawa al-Hindiyya; Ibn Abidin, Radd al-Muhtar]
The inward manners of supplication is exhibiting our neediness and absolute slavehood to Allah, consigning one's matters to Allah, and knowing that the only one who gives or benefits is Allah Most High. With this, one should be certain that Allah Most High answers our duas, but in the way He knows is best for us.
When circumstances do not permit us to pray 2 rakats, one should still turn to Allah in supplication, raising one’s needs to Him, for He answers our supplications and loves being asked