What is the authenticity of these narrations, and what is the ruling on supplicating with them with the intention of relief and the answering of prayers?

Reports transmitted from previous prophets, which have not been authenticated by the Messenger of Allah, peace and blessings be upon him, are among the reports of Banu Isra'il. These should neither be believed nor disbelieved except with legitimate Shar'i evidence. It is permissible to transmit them for consideration and admonition. As for supplicating with prayers mentioned in these reports, there is no harm in it, so long as their wording and meanings do not contradict the Shari'ah.