Does waking up for Fajr prayer before the Adhan count as waking up during the night, and is it permissible to recite the supplication of waking up (du'a al-ta'arr) at this time?

Waking up at night (تعار الليل) means waking up before dawn. It is prescribed for whoever wakes up at this time to recite the du'a (supplication) for waking up at night that is mentioned in the Hadith, and he is hoped to receive its reward.