Is it permissible to complete university studies after being accepted through nepotism and repenting of that, knowing there is fear of not being competent later on?

Not all intercession is forbidden. The intercession mentioned in the question is only prevented by the questioner's doubts and whispers [waswas]. One of the most beneficial treatments for whispers is to disregard them and not succumb to them. Ibn Hajar al-Haytami was asked about the ailment of whispers, and he replied that it has an effective remedy, which is to completely ignore them. He stated that if they are not paid attention to, they will disappear after a short time, but if one listens to them, they will only continue to increase. It is narrated in the Sahihain [Bukhari and Muslim] that whoever is afflicted with whispers should seek refuge in Allah and desist.

There is nothing in what the questioner mentioned that prevents her from pursuing her studies or permissible work, while observing the Sharia guidelines for a woman's going out, such as not wearing perfume or displaying adornment, avoiding sitting next to men and conversing with them unnecessarily or without need, and without softening her speech, and being constantly vigilant and mindful of Allah Almighty.