Is it permissible to work as a physical education and sports teacher in a secondary school where there is free mixing [between genders], knowing that entering this specialization was at the request of a professor of Islamic Sharia to protect the children from teachers who do not value religion and morals?

The basic principle is that it is not permissible to work in institutions where corruption and indecency are widespread, due to the exposure to temptation involved. However, since there is a religious interest in having an upright teacher who is trustworthy with Muslim children, there is no harm in your working as a physical education teacher, provided you ensure your own safety from temptation, adhere to the Sharia (definition) regulations in dealing with female students, enjoin good and forbid evil, and make your work a means for da'wah and advice.