"Ameryka", named after the Bristol Merchant Venturer who financed the expedition, was "discovered" by John Cabot in 1497 - he was looking for a sea-route to "Cathay" (China).
In 1498 he led a larger expedition from Bristol, which sailed as far South as Florida. (Henry VII wouldn' let him go further that way to avoid trouble with Spain - he was arranging a Spanish marriage for his son.)
This expedition established that the "new-found land" was NOT China - and a serious obstacle to getting there, so,in 1500, Sebastian Cabot tried to find a way around it by a "North-West Passage", but was stopped by ice. It also showed that there were no opportunities for trade there - no great civilization to trade with - or conquer and plunder, as in Central and Soth America - and so the area was "written off" as valueless..
70-odd years later, Ralegh and Grenville made three attempts to set up a colony on Roanoke Island
in Chesapeake Bay,but all failed.The war with Spain then itervened, and it was not until early in the next century that a permanent settlement was established at Jamestown in "Virginia", for commercial purposes, especially tobacco-growing. Then came the 1620 "Pilgrim Fathers",dumped much further North, but managing to survrve. They were NOT seeking "religious freedom" - but freedom for their OWN particular sect - not the same thing at all !)
As regards earlier "discoveries", probably the first European to do so was the early-mediaeval Irish monk, Brendan. The story of his voyage was thought to be mythology - until, in 1976, Tim Severin built an identical boat and sailed it there, with most of Brendan's observations coming up "on schedule" !