How can a Muslim rid himself of despair and recurring sins after failing in his studies, and return to acts of obedience?

Sins and transgressions have dire consequences in this world and the Hereafter. One must repent from them immediately to Allah Almighty. If a servant returns to a sin, they must repent again, and so on, every time they return to it, they must repent. They should beware of despairing of repentance after relapsing into sin, for repentance is accepted even if the sin is repeated.

As for failing an exam, one must exert effort in studying and learning, and seek help from Allah and rely upon Him. Failing does not mean the loss of this world or the Hereafter, nor does it mean the loss of sustenance, for sustenance is decreed. Therefore, failure should not lead one to disobey Allah Almighty.