If you didn’t get passing grades, have missed an important assignment, or you want to improve your grades for particular assignment/exam/test, it is not too late to salvage the situation. If you are worried enough you can convince your professor to consider giving a pass grade so that your future career does not get affected.
Here are a few tips to write an email to the professor to convince him/her to change your grades or to accept your apology for late submission of work.
Using a word Dear, respected etc. establishes that you view the relationship with your professor in a very professional way. Do not use words “Hi” or any other opening words which may be less formal.
Use honorifics such as professor and if he/she has a Ph.D., call them dr. this goes a long way because most students unknowingly poke on the professor’s ego by skipping the honorifics. Don’t call them by writing “Mr.” or “Mrs.” Also make sure that you spell their name correctly. Proof read multiple times before sending so that it is free from grammatical and spelling errors.
You can start off your first sentence with “I hope you had a nice weekend” “It is nice weather today” etc. Professors like it when people acknowledge that they have a life outside academics. It shows that you have a genuine interest in them. Rather than giving an impression to just get the grades for that time. Also, it’s not a good etiquette to simply ask for the favor alone, which could create a negative impression about you.
Tell them how they know you
It is unfair to expect to your professor to know you by your name since they might be handling at least hundreds of students. State that you are from this batch and you sit in this class of his. You could also add any information regarding personal meetings which you had with him earlier. This will help him/her remember about you to some extent, if not completely. It will be of immense help when you personally meet him after sending the email.
State the real reason for the email
You should have a proper subject title for your email. Please remember that this portion gives the main reason for sending the mail and hence do not go around about coming to the matter. Keep it simple and concise. Balance between being reasonable and being courteous. Don’t give too many excuses for failing in the grade.
Don’t give too excessive details and refrain from making demands. If you are finding it difficult to phrase the sentences, you can write that “I hope to meet you to talk about this subject. Can you specify a time for that?”
Prove that you are sincere in your studies
You can highlight the times when you have been able to perform good in his class. You can explain that you have taken a lot of steps to improve your grades. If there is sincerity in this, the professor can understand your difficulty. May be your grasp of the subject is less. If you explain your problem and that you are sincerely trying can earn brownie points for you.
Apologize for the poor performance
Give a background for the poor performance of you in the last exam. It may be various reasons such as being ill, underestimating the difficulty of the subject, improper planning, other personal issues etc. It is necessary to give the correct reason for the poor grades. Explain the reason in proper terms and avoid being vague. Give only the reasons which are authentic. If you have any proof that supports your absence or being ill, that can be attached to let the professor know about it.
Polite request for the favor
This is the moment you have been waiting for. Your entire life hinges on this part. Politely ask your professor to do a favor for you. Avoid being pushy or desperate. It is a request and your professor have the authority to reject your request. Don’t be needy too much. State the reason why you have been failing in your grades, it may be your health, or financial situation, or you have too much responsibility. Tell the professor that you would require one more chance to right the wrong.
Sign off properly
After you have stated your request, sign off with a simple thank you, or even better write “yours sincerely”. It’s a good gesture to express your gratitude to them. Now, you have done your job. Now it’s time to move on and hope for the best.
Sample Email asking for a favor (change of grade)
Subject: change of grades regarding
I hope you had a nice weekend last week.
I am a (year, major) at (university) and I am studying in your (subject) class. You may remember me as the one who sits in the first bench in your class. I have once answered your question about (question). As you know, I take my performance in your class seriously.
I had fallen ill last month, and I could not prepare for the exam. As a result, I have got C grade in the last exam. I sincerely apologize for the lack of preparation I had, and I regret my poor performance. As you know that I am disciplined and well-mannered student who diligently works for the grades, but I could not score good marks in that exam.
I request you to consider changing my grade taking into account my consistent past performance. I would be grateful for you if you can do this favor for me.
Sample apology Email for late submission
From: [Full Name]
Roll No: 
Batch NO: [XYZ]
Professor [Dr. Full Name]
[ABC] College of arts and science,
Dear sir/ madam,
Subject: requesting to accept my apology for late submission of [ABC] assignment – regarding.
I write this letter with great guilt that I have missed the deadline for submitting the assignment. It was supposed to be submitted before three days. I regret my behavior and sincerely apologize for the same. I regret that I have not planned my activities properly and did not foresee that I would lapse in submitting the assignment. I have got caught up in the cultural activities of the upcoming function in our department. I have been traveling outdoors to meet up with sponsors and doing fundraising.
But I have realized my mistake and have given my due diligence in completing the assignment which I have submitted along with this letter. Even though I had less time to complete it, I have ensured that the assignment meets the quality standard of yours and I am sure that you would find my efforts that have gone into preparing for this assignment.
I request you to accept my sincere apology and allow me to submit the assignment today. I assure you that this kind of lapses will not happen again. As I know that you take concerns about student’s academic grades, I promise you that I will excel in my grades and become a well-mannered student whom you can be proud of.
For more ideas on writing to your teacher, you can check sample apology letters to teacher.