There is no single supplication called 'the dua for a need' that is recited and the need is met. There are comprehensive supplications, and there are etiquettes which, when observed, make an answer more likely.
The best known here is narrated from Anas: 'O Allah, nothing is easy except what You make easy, and You make difficulty, if You will, easy.'
The etiquettes are four: begin with praise of Allah and salawat on His Prophet; persist without haste; seek out the moments when supplication is answered; and keep your earnings lawful — the Prophet ﷺ described a traveller, dishevelled and dusty, raising his hands to the sky while his food is unlawful, and asked: 'How can he be answered?'
اللَّهُمَّ لَا سَهْلَ إِلَّا مَا جَعَلْتَهُ سَهْلًا، وَأَنْتَ تَجْعَلُ الْحَزْنَ إِذَا شِئْتَ سَهْلًا
Allahumma la sahla illa ma ja'altahu sahla, wa anta taj'alul-hazna idha shi'ta sahla.
O Allah, nothing is easy except what You make easy, and You make what is hard, if You will, easy.
اللهم لا سهل إلا ما جعلته سهلا، وأنت تجعل الحزن إذا شئت سهلا
اللَّهُمَّ إِنِّي أَسْأَلُكَ مِنْ فَضْلِكَ وَرَحْمَتِكَ، فَإِنَّهُ لَا يَمْلِكُهَا إِلَّا أَنْتَ
Allahumma inni as'aluka min fadlika wa rahmatik, fa-innahu la yamlikuha illa ant.
O Allah, I ask You of Your bounty and Your mercy, for none possesses them but You.
اللهم إني أسألك من فضلك ورحمتك، فإنه لا يملكها إلا أنت
يَا حَيُّ يَا قَيُّومُ بِرَحْمَتِكَ أَسْتَغِيثُ، أَصْلِحْ لِي شَأْنِي كُلَّهُ، وَلَا تَكِلْنِي إِلَىٰ نَفْسِي طَرْفَةَ عَيْنٍ
Ya Hayyu ya Qayyum, birahmatika astaghith, aslih li sha'ni kullah, wa la takilni ila nafsi tarfata 'ayn.
O Ever-Living, O Sustainer, by Your mercy I seek relief. Set right all my affairs and do not entrust me to myself for the blink of an eye.
يا حي يا قيوم برحمتك أستغيث، أصلح لي شأني كله، ولا تكلني إلى نفسي طرفة عين
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'Allahumma la sahla illa ma ja'altahu sahla, wa anta taj'alul-hazna idha shi'ta sahla' (Ibn Hibban).
Because the delay may itself be better; because the answer may come as an evil averted or as a reward stored for the Hereafter; and because standing longer at His door is itself a gift.
