First of all, it should be clear that reading the Glorious Qur’an entails great rewards. The Prophet, peace and blessings be upon him, is reported to have said: “Whoever reads one letter of the Book of Allah, will have one Hasanah (reward) for doing so, and every Hasanah will be multiplied by ten. I do not say that ‘Alif, lam, mim’ is one letter, but ‘Alif’ is a letter, ‘lam’ is a letter and ‘mim’ is a letter.’”
Coming to the point raised in the question, we would like to state that covering the `Awrah when reading the Glorious Qur’an is not obligatory, but a Muslim is always recommended to be in a state of purity or Taharah and modesty when listening or reading the Qur’an.
Dr. Marawan Shahin, Professor of Hadith and its Sciences, Faculty of Usul Ad-Din (Islamic Theology), Al-Azhar Univ., states: “Though it’s not a condition to cover the `Awrah while reading the Qur’an, it’s far better and recommended to read the Qur’an while one is in the best form of dignity and modesty, for the Qur’an is the Words of Allah, Most High. Therefore, covering the `Awrah while reading the Qur’an is recommended, though not obligatory.”