How to politely request payment for extra hours of TA service rendered?
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.