Could you help me correct my writing? Thank you very much!

Dear Mr. Smith,
I am a graduate majoring in Public Administration. The purpose of <font style="background-color: rgb(255, 255, 0);">the</font> letter is to apply for the position as a secretary.

                                                                                                            ^^^ For the way you've worded your sentence, <font style="background-color: rgb(255, 255, 0);">"this"</font> would be more appropriate, as it just sounds awkward otherwise

I believe I am quite fit for the post. <font style="background-color: rgb(255, 255, 0);">Firstly</font>, not only does my professional knowledge matches this vocation well, but also my English

                                                            ^^^ The same as the last correction, it is simply an awkward placement of "Firstly" so just shorten it to <font style="background-color: rgb(255, 255, 0);">"First"</font>

speaking is proficient enough to deal with the foreign receptions. Second, as a detail-oriented and patient person, I could easily accommodate myself to this job. Lastly, regular exercises and optimistic state of mind would help me manage work stress better.
My resume is enclosed with the letter. It would be appreciated if you could grant me an interview.

Overall, Good Job! I advise you just get a bit more time talking to native English Speakers