（中文是想要表达的意思，翻译不必和中文完全一样）
中文部分：
附件是我给Professor 'XXX'（某某科目的coordinator）写的最后一封信，在信里，我解释了为什么这个科目对我是如此的重要并且向他申请了补考的机会。
我的翻译：
Attached is last letter which I email to Professor 'XXX' who is the coordinator of 'XXX'，In this letter, I explained why this subject is so important for me and apply for a make-up examination:
Attached is the latest email that I sent to Professor 'XXX', the coordinator of 'XXX', in which, I explained why this subject is so important for me and applied for a make-up test.
"latest" is better than "last", and you don't say "email a letter" often
if you want you could use one long sentence
"applied" instead of "apply"
"make-up exam/test" is OK but