WASHINGTON—The Biden administration on Thursday said it would immediately begin turning away Cubans, Haitians and Nicaraguans who cross the U.S.-Mexico border illegally, a major expansion of an existing effort to stop Venezuelans attempting to enter the U.S. Instead, the administration will accept 30,000 people per month from the four nations for two years and offer the ability to legally work, as long as they come legally, have eligible sponsors and pass vetting and background checks.