I don't know whether it's common in the UK (or in the USA or any other English speaking country) but in our country, if a student was absent at the previous lesson, he or she says "I'd like to apologize. I was absent last week." => the teacher isn't going to examine the student's knowledge of the theme that was taught at the last lesson.
My question is very simple - I'd like to know whether the verb "to apologize" can be used in such a meaning I've just described.
Apologize is defined as to tell someone that you are sorry for having done something that has caused them inconvenience or unhappiness - but in the example I've described, the student is not too sorry and doesn't cause the teacher any inconvenience (we can say).
Well, the point is this. English isn't part and parcel her culture(s). If in your language apologize is used in that context, then use should use it that way. Why not?
To answer your question, some teachers, no matter their country of origin (be it on North America, Australian, New Zealand, or UK soil) would consider the student's apology a matter of politeness. However, the teacher wouldn't necessarily get the pragmatics there. That apologizing actually means don't expect me to know the previous lesson's material. Quite the opposite actually. In North American, which is where I am from, we kind of expect the student, absent--or dead, to have done the reading for each lesson, prior to stepping into the class. Aplogizing is a formality. Nothing else.