Hi Andrew In answer to your question, I would say that the ideal translation is "Examén de Recuperación" or "Exámen de Regularización" in the second sentence, i think is more common use this sentence in case your students are reprobate in their lessons

YOU CAN SAY , EXÁMEN DE RECUPERACIÓN OR JUST RECUPERACIÓN.

Hi Andrew! You could say: "Examen Recuperativo" Follow me and we can practice your Spanish and my English. Saludos