I forgot to enter my middle name on my FLR (M) application. I just wrote up a cover letter, signed it, and uploaded it with my documents. I also took it to the biometrics appointment in case it didn't get properly uploaded. I made sure to mention my error at my biometrics as well, to be sure all bases were covered.
I did end up having my BRP arrive without my middle name so I had to get a correction made on it. I had no issues, just had to wait for a replacement BRP. I don't think it held up my application either. I also had no issues obtaining subsequent ILR, citizenship, and passport.
Best of luck!
Found this from my post history, you can see I had similar issues and worries:
https://talk.uk-yankee.com/index.php?topic=98165.msg1299308#msg1299308