Is it permissible to use influence to obtain a university seat if the competitions are not fair, and is the income earned from a degree obtained in this way lawful or unlawful?

There is no harm in using influence to repel injustice and achieve what one deserves. However, it is not permissible to use it to obtain what one does not deserve, as that involves deception, fraud, and injustice to others. Whoever obtains a position they do not deserve commits a sin by doing so. Nevertheless, the wages they earn for a permissible job obtained with this certificate are permissible.