Which is less bad: praying at home at the time of the Adhan, or praying late at the mosque (in congregation or alone)?

If you hear the Azan and your going out for prayer will not result in harm, then you must go out to perform it in congregation at the mosque. However, if your going out would cause significant harm, such as losing your job, then you may pray with the students inside the classroom or in a clean place. If you are unable to do so, then delay the prayer to perform it in congregation if you are certain that a congregation will be available at the end of the time, otherwise pray individually at the beginning of the time.