I don’t think you make another Bracha, unless you have a significant Hefsek. You’ll have to consult LOR for verification and details.
If a Ner burned out you have two things going for you. First of all, you are already Yotze with the first one. Secondly, the Halacha is that as long as you lit it well in a way that it should burn for the half hour, then even if it goes out you were Yotze. We are Noheg to relight it but that is not the basic Halacha.
You are not allowed to put the candles out before the half hour. Putting them out to relight a candle would be doing what you shouldn’t do, without any gain.