レッスン開始に遅れてしまいすみませんって英語でなんて言うの？

I apologize for my tardiness/lateness - "I apologize" is a fancier way to say "I'm sorry" and adding "sincerely" stresses that you are "truly sorry" I am so deeply sorry. - this is one way to stress that you are really sorry My lateness is inexcusable - this tells the listener that even though you are about to give an excuse, you understand that there is no excuse I promise it won't happen again! - some people say that saying sorry means that you will try your hardest not to do something again; to show that you are truly sorry, you can add "I promise it won't let it happen again!" This means you won't allow the same thing to happen, and you will try to prevent it from happening as much as you can.

*I am sorry for coming late for the lesson. I had an internet problem. This is a simple way of apologizing and it is polite. *Forgive my tardiness, I had internet problems. Tardiness means lateness. For example: "Forgive my tardiness, I had some very important business to attend to."

if a mechanical or electrical system or appliance 'plays up' then there is some kind of malfunction occurring. It may not be a continuous fault - it could be intermittent. You could also describe such an item as having a 'hiccup'. A hiccup is an undefined problem which has interrupted the smooth working of something. A "No coffee today?" B "Sorry, there's a hiccup with the coffee making machine."

Both of these phrases are accurate and will function as required although "I was late because my internet was acting up" is a more colloquial way of saying it. The phrase "acting up" is an english verbal phrase which means "to misbehave or not do as it is supposed to". I hope this helps

"I had internet/connections problems/issues that made me late to class." explains to your teacher that you were late for class because of problems you had with the internet.