The first attempt by the film mogul Harvey Weinstein at producing a theatrical show, a new musical called “Finding Neverland”, will begin preview performances on Broadway in March at the Lunt-Fontanne Theater and open on April 8. Adapted from the 2004 film about J.M. Barrie and the creation of his early-20th-century play “Peter Pan,” “Finding Neverland” broke box-office and attendance records this summer at the American Repertory Theater in Cambridge, Mass., where it finished two months of performances on Sunday. The show received mixed reviews. Mr. Weinstein has spent roughly $3 million to enhance the budget of the Cambridge production, and is expected to spend an additional $11 million to mount the show in New York – on the medium-to-high-end of budgets for new Broadway musicals.