If you are calling in sick (you are at home and want to inform of your absence) you would speak to the relevant person (in England, if you are at university you are probably on first name terms with your professor) and simply say "Unfortunately I am sick today and am unable to make it in" possibly your professor will ask what is wrong. As for asking to go home, similarly, you would say "I feel unwell and I think I am going to have to go home." This kind of setting is, I am assuming, a university setting. For school children it is different: their parents would report their absence from school in the first instance, and in the second a child may declare that they feel ill, and probably a teacher would decide whether they should be sent home or not.

The example looks okay. However, for English-speaking countries, it is recommended to address the surname of the teacher together with salutation rather than plainly calling them "Teacher". For instance, A) If the student has not yet arrived at school and called the teacher by phone, "Good morning, Dr./Mr./Mrs./Miss ... I don't feel well today, and thus I would like to take a leave so as to see a doctor/to visit a medical clinic." B) If the student has already arrived at school, but they got sick during the school and spoke to the teacher for leave, "Hi, Dr./Mr./Mrs./Miss ... I don't feel well now. I would like to leave now to see a doctor/to visit a medical clinic." === I wish you a good day.／祝　是日安好。