How to politely request payment for extra hours of TA service rendered? My TA letter indicates 90 hours working. However, I worked for 120 hours due to the volume of the students. Now my work is finished I want to let them know by writing an email to the professor. I want to be nice and thankful for the given position.  

Note, my goal is to get paid for these 30 hours. I had already raised the issue with the professor before doing the work, and was told to do the work, record my hours, and then the professor would follow up with the head of department.
Your letter seems perfectly polite. My one critique is that I'm not clear what you're asking for. Just to get your record updated? A larger assignment going forward? More money for the hours you put in already? I assume the last one (getting paid for those 30 hours), but I'm not sure. If you're struggling for words, you could call this a "stipend adjustment" or something like that.

It is good that you got approval before working these hours. If you had not (as the comments and other answers have noted), you would have very little chance of getting retroactively compensated. Hopefully you got a clear commitment in writing: there is still a chance this will be messy if the professor does not or cannot follow through on their promise to find extra funding for you.