In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
Answering your question, the late prominent Saudi Muslim scholar, Sheikh Ibn `Uthaymeen, stated:
There is no such thing as “Prayer of Need” or “Prayer for Memorizing the Quran.” This is because these kinds of acts of worship can only be confirmed through legal evidences. Neither of these have any evidences for them that can be considered proofs and acceptable in Islamic Law. Therefore, they are not sanctioned by Islamic law.
Salat al-Hajah is mentioned in four hadiths, two of which are fabricated. In one hadith, Salat al-Hajah has twelve rak`ahs and in the other it has two. The third hadith is very weak and the fourth hadith is weak.
Almighty Allah knows best.
Source: Islamic Fatawa Regarding Women, Compiled by Muhammad Ibn Abdul-Aziz Al-Musnad and translated by Jamal Al-Din Zarabozo, Darussalam, 1996.