All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
We do not know of a legislated prayer for the one in distress. Rather, there are supplications in the Sunnah that are to be recited at times of distress.
For instance, Abu Bakrah, may Allah be pleased with him, narrated that the Prophet, sallallahu ‘alayhi wa sallam, said, "The supplications of the one in distress are: 'Allahumma rahmataka arjoo, fala takilni ila nafsi tarfata ‘ayn, wa aslih li sha'ni kullahu, la ilaaha illa ant (which means: O Allah, it is Your mercy that I hope for, so do not leave me in charge of my affairs even for the blink of an eye, and rectify for me all of my affairs. None has the right to be worshiped except You)." [Al-Bukhaari in Al-Adab Al-Mufrad, Ibn Hibbaan, and An-Nasaa'i in As-Sunan Al-Kubra]
Ibn ‘Abbaas, may Allah be pleased with them, narrated that the Messenger of Allah, sallallahu ‘alayhi wa sallam, when he was in distress, used to say, "La ilaaha illa Allah Al-‘Atheem Al-Haleem; la ilaaha illa Allah rabbul-‘Arshi Al-‘Atheem, la ilaaha illa Allah rabbu-Samaawaati wa rabbu Al-Ardhi wa rabbul-‘Arsh Al-Kareem (which means: none is worthy of worship but Allah, the Great, the Forbearing; none is worthy of worship but Allah, the Lord of the Great Throne, none is worthy of worship but Allah, the Lord of the heavens and the Lord of the earth, and the Lord of the Noble Throne)." [Al-Bukhaari and Muslim] This wording was cited in Saheeh Muslim.
It has been narrated on the authority of Asmaa' bint ‘Umays, may Allah be pleased with her, that the Messenger of Allah, sallallahu ‘alayhi wa sallam, said to her, "May I not teach you phrases which you utter in distress? (These are:) 'Allahu, Allahu rabbi la ushriku bihi shay'an.' (Which means: Allah, Allah is my Lord, I do not associate anything as partner with Him.)" [Abu Daawood, Ibn Maajah, and others]
Sa‘d, may Allah be pleased with him, reported that the Prophet, sallallahu ‘alayhi wa sallam, said, "The supplication of Thun-Noon (Prophet Yunus [Jonah], may Allah exalt his mention) when he supplicated while in the belly of the whale was, 'La ilaaha illa anta subhaanaka inni kuntu min Ath-Thaalimeen.' (Which means: There is none worthy of worship except You, Glory to You, Indeed, I have been of the transgressors.) Indeed, no Muslim man supplicates with it for anything, ever, except that Allah responds to him." [Ahmad and At-Tirmithi]
It is also prescribed to perform a two-Rak‘ah (units of prayer) prayer of need (Salaat Al-Haajah), and they are not particularly prescribed for distress. Please refer to fatwas 3749 and 82384 about the prayer of need.
Allah knows best.