Du’aa’ whilst reading Qur'aan
This is permissible and was done by the Prophet (blessings and peace of Allah be upon him). When he prayed tahajjud at night, when he came to a verse of warning he would seek refuge with Allah, and when he came to a verse of mercy he would ask his Lord for mercy. So there is nothing wrong with that; rather it is mustahabb when praying tahajjud at night and when praying by day, or when reading the Qur'aan outside of prayer. All of that is mustahabb and you do not have to repeat the Basmalah or words seeking refuge with Allah. Rather you may offer this du’aa’ then start reading again, without needing to repeat the words seeking refuge with Allah or the basmalah. All of this applies if you are praying alone in a naafil prayer.
But if you are offering an obligatory prayer, it is not proven that the Prophet (blessings and peace of Allah be upon him) used to do this in the obligatory prayers. So if you are praying with the imam, you should listen attentively to your imam and not offer these du’aa’s when the imam is reciting. Rather you should remain quiet and listen attentively in the prayers in which Qur'aan is recited out loud. In the prayers in which the recitation is silent, you should recite al-Faatihah and whatever else you can, without saying the du’aa’s that you say in the naafil prayers, because the Messenger (blessings and peace of Allah be upon him) did not do that in the obligatory prayers. Perhaps the reason for that -- and Allah knows best -- is to make it easier for people and not make the prayers too lengthy, because if he were to offer du’aa’ at every verse which mentions mercy and seek refuge with Allah at every verse in which there was a warning, the prayer might become too long and might be too difficult for people. It is by Allah's mercy and His kindness towards His slaves that He did not prescribe that in the obligatory prayers so that the recitation would be uninterrupted and so that the recitation would not be too long for the people. But in the naafil prayers or in tahajjud, or at night or in Duha prayer and other naafil prayers, there is nothing wrong with doing that and the matter is broad in scope. End quote.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him).