Most apologies that fail in professional life do not fail because the words were wrong. They fail because the apology was offered as a transaction, and the other person was waiting for a relationship.
A good apology names the specific thing, acknowledges the cost to the other person rather than the cost to you, and does not arrive bundled with an explanation. The explanation can come later, if it is invited.
What is left unsaid matters as much as what is said. No self-defence. No reference to your own difficult week. No early move toward what happens next. Just the recognition that something landed, and that you are the one who placed it.
When the apology lands, you will usually feel the room change. Not lighter exactly. Steadier. That is the sound of a relationship being given back its weight.