Harvard University’s endowment plans to lay off 57 people beginning in April as it seeks to cut its 230-person staff in half this year. Harvard Management Co. said the first separations will begin April 18, according to its filing this month with the Massachusetts Department of Labor. Emily Guadagnoli, a spokeswoman for Harvard Management, declined to comment. Other staffers are poised to join funds that will be spun off through this year as internal investing operations are curtailed.