Why did Allah permit Satan to whisper to the children of Adam, to know what they are thinking, and to cause them to doubt their prayers?

It was Allah's wisdom to make humankind His vicegerents on Earth and to test them by giving Satan power over them, in order to distinguish the wicked from the good and to reveal who obeys Allah and who responds to Satan. Satan gains power over those who align with him and associate partners with Allah, whereas the sincere ones are protected by Allah from his schemes.

The creation of Iblis (Satan) entails many wisdoms and benefits, including: Completing the ranks of servitude for prophets and saints through struggling against Satan and seeking refuge in Allah from him. Instilling fear of sin among angels and believers after witnessing Iblis's fall. Making Iblis a lesson for whoever disobeys Allah's command and becomes arrogant, just as Adam's sin was a lesson for whoever repents and returns. Distinguishing the wicked from the good among humans, as Iblis is a touchstone that discerns this. Manifesting Allah's perfect power in creating opposites, such as Iblis and the angels. Highlighting the beauty of a thing through its opposite, for the ugly reveals the virtue of the beautiful. Achieving gratitude in its various forms from the saints due to the existence of Iblis and their trial by him. Demonstrating Allah's love for acts of worship that can only be achieved through struggle and self-sacrifice, such as love, repentance, reliance [on Allah], patience, and contentment. Revealing Allah's signs and the wonders of His power and the subtleties of His creation in confronting those who oppose His messengers. Manifesting the effects of Allah's beautiful names and attributes, for the existence of transgressors allows for the display of forgiveness, pardon, forbearance, and justice.