In the movie Dawn of Justice, Batman is chasing a group of people taking the Kryptonite to Lex. Superman intercepts him and foils Batman's plan. The Kryptonite is being taken in a regular wooden box. Why does Superman not notice that it is Kryptonite that is being transported and fought over? Had he seen it, he might have acted differently? For him, wouldn't he rather no one got their hands on the Kryptonite?
This is the first time kryptonite is introduced in this universe. At that point the only people who knew kryptonite can effect Kryptonians is Bruce Wayne and Lex Luthor (and others working with the two).
In Man of Steel a prequel to Batman v Superman, Superman explains he learned to hone his ability and control his senses.
The movie is consistent in covered kryptonite does not appear to hurt Superman. So if he can control his powers and kept focus on Batman, it would not make sense for him to detract his attention on a rock he does not know can hurt him.
In this movie Superman also misses the bomb in the congressional hearing and comments he should have been more aware. Superman was also totally focused on Batman at this stage and trying to stop him doing his Bat thing. Superman, while not being dumb, is not particularly known for being as intelligent as many, many people in general, even Brue Wayne for instance (Superman doesn't invent anything, look at all the amazing things Alfred and Bruce Wayne invent).
So where other characters may have thought "why is Batman doing this? Who is he chasing?" Superman has not really thought through his decision to stop Batman and forgets all the good the Bat has done in the past. Superman fixates on the fact that Batman effectively condoned murder by branding the sex traffickers with the Bat sign.
All in all this misstep by Superman is very much in character with everything else Superman does in this film. I mean he is so arrogant and sure of himself he assumes Batman can not hurt him and gets his arse handed to him by the Bat. Poor cognitive powers, Superman.
I can relate.
Scene 1: When Lex's people are taking kryptonite from the docks. It was in the black box inside the wooden box. The black box might be made up of lead so it does not spread the radiation. If it does not spread the radiation then superman wouldn't know about its presence(my opinion).
Scene 2: When Lex forces Superman to fight Batman just before doomsday comes in action. The batman carried bombs and spear made up of kryptonite. At that time also Superman didn't knew about kryptonite's presence with Batman. May be because batman had it covered with lead shells and spear was stuck in land in the building. So that Superman might miss kryptonite and couldn't take action on kryptonite before the fight with batman.