Supplicating between the two prostrations is legislated by the Shafi’s and the Hanbali. The Hanbali have made it obligatory, and the least one can say is “My lord Forgive me” as it was reported of the Prophet ﷺ
Dr Wahba Al Zuhaili -May Allah have mercy on him- in his book of Islamic Jurisprudence and its evidence says:
There is no set supplication between the two prostrations for the Hanafis, like there is no supplication after rising from Ruku’. There is no supplication in Ruku’ or Sujood according to the Hanafi school of though as we have presented, what has been reported is said during the voluntary prayers (nafl) or during Tahajjud.
The Malikis have not mentioned that supplication is part of the requirements for prayer, and Ibn Jizi mentioned about what is said between the two prostrations (sujood)
Supplication is recommended by the Shafi and the Hanbalis. The Hnabalis said: It is obligatory and the least one should say is (My Lord forgive me) once, and the lowest complete for them is saying it three times, like what is said in the Ruku and Sujood.
The wording for this supplication for the Shafi, Maliki, and Hanbali, is: (My lord forgive me, have mercy on me, strengthen me, raise my rank, sustain me, guide me, and pardon me). The Hanbalis have said: It is not allowed in the prayer, other than what is mentioned in the sunnah, And it is not allowed to supplicate for things not about the afterlife, such as the needs of this world and its indulgences, and the prayers is invalidated by them.
The hadith which legislates this is what was narrated by Abu Hudhayfah: “That he prayed with the Prophet ﷺ and he would say between the two sujoods: My Lord Forgive me”
It is narrated by Ibn Abbas that he said: “The prophet ﷺ would say between the two sujood: Oh Allah Forgive me, have mercy on me, guide me, pardon me, sustain me”
And in a narration of Muslim: “A man came to the Prophet ﷺ and he asked: What do say when I ask my lord, He said: say: Oh allah Forgive me, have mercy on me, sustain me, because all these join together your worldly life, and your afterlife” this is because forgiveness conceals you, and pardoning: protects the person from trials, and there are two types of sustenance: Visible to the eyes, such as power and strength, and what cannot be seen such as the things in the heart and soul such as knowledge and sciences.