The main reason for Henry Hudson's voyage was to find a northwest passage to Asia. At the time, there were only two known routes to Asia: the long and dangerous journey around the Cape of Good Hope, and the even more dangerous journey through the Arctic ice. Hudson believed that there must be a shorter and safer route through the Americas, and he set out to find it.
To claim new land for England
England was a rising power in the 17th century, and it was eager to expand its territory. Hudson's voyage was also seen as an opportunity to claim new land for England. If Hudson could find a northwest passage to Asia, he would also open up a new trade route to the East, which would benefit England's economy.
To gain fame and fortune
Hudson was also motivated by a desire for fame and fortune. He knew that if he could find a northwest passage, he would be hailed as a hero and would be rewarded handsomely by the English government.