What is the authenticity of the hadith narrated on the authority of Aisha, may Allah be pleased with her, concerning her dream in which she was taught a supplication to relieve her distress, and what are the authentic supplications for relieving the distress of the oppressed?

The tradition mentioned in the question is found in Ibn al-Najjar's Tarikh Baghdad and was cited by Al-Suyuti and Al-Alusi. Ibn al-Najjar and Al-Alusi described it as strange. Scholars have not ruled on its authenticity or weakness. There is no objection to supplicating with it, even if it is weak, because the wording of supplications is not tawqifi (restricted to what is prescribed), and because many scholars permit acting upon a weak hadith in matters of fada'il al-a'mal (virtues of deeds).

There are other authentic supplications, such as reciting "Qul Huwa Allahu Ahad" (Say: He is Allah, [the] One) and the two Mu'awwidhatayn (chapters of refuge) three times in the morning and evening, which are sufficient for everything. Among the authentic supplications is also the supplication of Ibn Mas'ud (may Allah be pleased with him) when fearing injustice or tyranny from a ruler: "O Allah, Lord of the seven heavens and Lord of the Magnificent Throne, be my protector from so-and-so the son of so-and-so and his partisans among Your creation, lest any of them should transgress against me or act tyrannically. Glorified is Your protection, and exalted is Your praise, and there is no god but You."