To what extent is it permissible to recite the Holy Quran while one's awrah is exposed, or while a woman has her hair uncovered?

It is not a condition to cover one's awrah (private parts) for reading the Qur’an. However, it is desirable for a Muslim to cover their awrah and be in a presentable state, because they are communing with Allah. Abu al-Aliyah used to wear a turban, dress well, and face the Qiblah when reciting. It was said in this regard: "Prepare for the intimate communion with the King, the Requiter, by purifying yourself from ritual impurity and purifying the place, and listen attentively to Him, wear your best clothes, and maintain dignity and tranquility."