Is what the student did considered cheating, and does his repentance erase the effect of this cheating, and how can he free himself from its consequences?

The teacher's action of erasing what the student wrote after the time was up indicates that writing an answer after the time limit is a prohibited form of cheating, based on the Prophet's (peace be upon him) saying: "Whoever cheats us is not of us." Repentance from cheating is obligatory, and the points obtained through it are not permissible. If these points will give the student an advantage over their closest competitors in obtaining the scholarship, then the scholarship is not permissible for them. They must inform the responsible party about the truth of their cheating, or leave the scholarship for whoever deserves it. The student is advised to rely on Allah and strive in their studies so that their success is significantly greater than their closest competitors, making the scholarship permissible for them.