Apologizing sincerely

Lo siento mucho

loh see-EN-toh MOO-choh — I'm very sorry

"Lo siento mucho, llegué tarde por el tráfico."
I'm very sorry, I was late because of traffic.

"Lo siento" on its own is fine for small things; adding "mucho" (or "muchísimo" for even more emphasis) is what you reach for when you actually mean it.

"What I say, genuinely, whenever I have to reschedule a lesson last minute."— Graciela