我会实事求是地写：

邮件发出后再重读一遍时发现我拼错了你名字，which I didn't notice during my proof reading. I sincerely apologize for the mistake. 然后再加点为他们工作的意向。。。Overall, make the note short and sweet.

However, if you don't feel comfortable to send the second email, follow your intuition.

Good luck to you.