All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
It is permissible to supplicate during the times when the prayer is disliked or prohibited. It is the prayer that is disliked at those times and not the supplications; rather, the mention of Allaah and supplications are desirable during this time as it is the time of the legislated morning mention of Allaah. The Prophet said: "Whoever prays Fajr in congregation and sits mentioning Allaah until sunrise, then prays two Raka'h, he will get a complete reward of Hajj and 'Umrah, (he repeated 'complete reward' thrice)." [At-Tirmithi]
Finally, it should be noted that the most preponderant opinion about the Night of Decree is that it is not definite and constant, but it is an odd numbered night during the last ten days of Ramadan. The night ends by the commencing of the time of dawn. For more details on the best times of supplications, please refer to Fatwa 88296.
Allaah Knows best.