Presumably "answering to your section head" in this context means that you are initiating the correspondence to explain and apologize, rather than answering a specific communication from the section head. This is obviously a difficult situation, and the tone and content of your message will depend to some extent on the workplace culture. However, I would recommend the following strategies to make it as effective as possible.
First, give a clear, unambiguous apology in the first sentence of the message. Having done this, do not keep repeating the apology. Saying sorry ten times in a message is no more effective than saying it once, in fact it is less effective, since it generally becomes irritating. Groveling is not only undignified, but counterproductive. More generally, do not ramble. Keep your message concise.
Once you have apologized, give any excuses or extenuating circumstances in the first paragraph. Again, do not repeat them and generally keep the tone neutral and professional, without emotive language or pleas for clemency.
Try to think of an effective way in which you can go above and beyond the call of duty to make up for this error. If no obvious way presents itself, ask your section head if there is anything you can do to make his/her life easier and prove how seriously you take your job.
This may be the most controversial point, but I would request an early, brief meeting with the section head, specifically to find out what you can do to be particularly helpful going forward. This reframes what might have been labeled a "punishment" as a voluntary task that you willingly perform.
After the apology and explanation/excuse, the rest of your message, which should not be more than three short paragraphs, should be focused on the future and what you can do to be most helpful.