I rewatched Children of Men to watch the long takes and noticed blood on the lens of the battle scene at the end and I'm wondering if it was intentional or not. I'm thinking it happened on accident and they decided to go with it. Either way, a gutsy call.

It was not intentional i believe however when they played the footage back and saw it, the DoP liked it and the gritty realism it added to the shot so they kept it. Thats not the exact story but close anyway :)

What happened was, the director saw it on his monitor and yelled cut, but the DOP couldn't hear him, so they kept on shooting. It wasn't intentional.
But afterwards they liked the effect although they digitally removed the blood to the end of the shot, when Theo goes inside the building, because they felt then it was becoming distracting.