I need you to write a letter to one member of the committee

I need you to write a letter to one member of the committee. In general the Doctoral comprehensive exam consists of 3 areas, I failed at one area (Operating System ) and i need you to write a letter for her. I want to tell her that I applied for a petition to the committee and because she is one of the committee members I need her support.

She is the responsible for Operating System .The goal from this letter to tell her please support me, I want to stay in PhD program and I am ready for any request. Please rewrite the letter (attached file) to make it strong without any grammar mistakes.