Is Salat al-Hajah authentically reported from the Prophet (PBUH)? And if so, how is it performed?

Scholars have differed on the ruling concerning Salatul Hajah (the Prayer of Need) due to their disagreement on the authenticity of its hadith. Some opine that it is impermissible to act upon it due to its weak chain of narration, as it contains a narrator whose hadith is abandoned (matruk al-hadith). Others hold that it is permissible to act upon it because it has various (chains of transmission) and (corroborating reports) that strengthen it. Furthermore, since it pertains to virtues of deeds (fada'il al-a'mal), a weak hadith may be acted upon in this context if it falls under an established principle and is not contradicted by something more authentic. This latter opinion is the more correct one, and it is held by a group of scholars. The manner of performing it is as described in the hadith.