What were verrazano, cartier, and hudson searching for in the americas?

They sought a northwest passage that would allow them to get to Asia more quickly. Due to the terrain and the transportation of the time, it was not practical for Europeans to use a land route to get to Asia, which was viewed as an area full of riches and exotic goods. However, the sea route to Asia was much longer than they could easily traverse. North America was originally discovered on an expedition seeking to find a shorter route to Asia by going in the other direction (they knew the earth was round at this point). Once America was discovered, it was only seen as a barrier to Asia. The northwest passage was a theoretical ice-free route around the Americas to the north that would allow a shorter trip to Asia. No northwest passage was ever discovered, and none really exists.