我會實事求是地寫：

郵件發出後再重讀一遍時發現我拚錯了你名字，which I didn't notice during my proof reading. I sincerely apologize for the mistake. 然後再加點為他們工作的意向。。。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.