First of all, we’d like to state that reciting a Qur’anic Surah or some Qur’anic verses after Al-Fatihah is a Sunnah in the first two rak`ahs (Prayer units) of any Prayer whether obligatory or supererogatory such as tahajjud Prayer.
However, if one is unable to recite any verse(s) or Surah after reciting Al-Fatihah, then his Prayer remains valid.

The eminent Muslim scholar and renowned Da`iyah, Sheikh `Abdel Khaliq Hasan Ash-Shareef, states: “Tahajjud Prayer is Sunnah which a Muslim is encouraged to follow all the time; it’s not something confined to Ramadan. In this Prayer, one is allowed to read after Al-Fatihah any surah or some Qur’anic verses according to his ability. Reciting Surah after Al-Fatihah is a controversial issue among Muslim jurists. The most correct view is that it is sunnah which means that the Prayer will be considered as valid if you read only Al-Fatihah in each rak`ah. However, a Muslim is encouraged to abide by the sunnah of the Prophet (peace and blessings be upon him) who encouraged Muslims to read as much as they can from the Qur’anic verses after Al-Fatihah seeking Allah’s great reward especially in this great month of Ramadan. May Allah reward you and accept your fasting and all your good deeds.