Yes, salat-ul-Hajah is an authentic salah. Whenever one is in need, one can offer this salah and supplicate Allah. It is narrated by Abdullah Ibn Aufa (may Allah be pleased with him) that the Messenger of Allah (may peace and blessings of Allah be upon him) said: “Whoever has some need from Allah or from any of His creation, let him perform ablution and pray two rak’ah, and praise Allah well after salah and send durood on the holy Prophet Muhammad (peace be upon him) then let him say the following dua:
لا إله إلا الله الحليم الكريم سبحان الله العرش العظيم ……إلى………يا أرحم الراحمين.
A narrator Faid Ibn Abdur Rahman has been termed weak in this hadith, yet Imam Tirmidhi has classified it as Hasan Gharib, thus he writes: this hadith is Hasan Gharib. (Sunan Tirmidhi, Hadith Number: 479) So, this narration is practicable.
Allah knows Best!
Darul Uloom Deoband