Account for the relationship as it is now
Use a tone that fits the present relationship, not the closeness, authority, or access that existed in the past. For a message about how to apologize for making an insensitive joke, use this factual base: name the exact action, the inconvenience it caused, and what you understand now.
Ask without assigning an obligation
Explain what you hope for while making decline, delay, or a different format genuinely acceptable. Offer the smallest useful repair and give a realistic time for completing it. Make the immediate next step clear enough that the recipient can answer without reconstructing the entire history.
Keep the ending honest
Do not inflate the apology, fish for reassurance, or demand immediate forgiveness. If the person has asked for no contact, repair what you can without sending another message. Avoid future promises, guilt, flattery, or nostalgia that imply the recipient owes you access.