No, it has not. It is a common misconception and an easy mistake to make.

Our sun is a second, if not third generation star. That means the hydrogen it is burning comes from an earlier star that, supposedly, had used up out all its hydrogen and exploded in a supernova. How is that possible?

Easy: the earlier star did not use up all its hydrogen. Only the hydrogen taking part in the nuclear reaction. Which is only a small part in the star's core. The most of it in the stellar body and atmosphere would have remained untouched. The shock of the supernova explosion would have turned some of that to heavier elements but the vast majority remained for the next generation star.

Our sun does not have enough oomph to become a supernova which means it will last longer than its predecessor(s) but it will still not use up all its hydrogen before its candle goes out.