Marcelo Claure proved himself adept at building a business from scratch, having founded a wireless distribution company, called Brightstar, in 1997 and building it into a multibillion-dollar enterprise.

Now, he has to prove himself as Mr. Fix It. In August 2014, Mr. Claure left Brightstar to become chief executive of Sprint Corp. Sprint has been in turnaround mode since a disastrous 2005 merger with Nextel. It has lost subscribers and money for years. Last year, Sprint abandoned plans for a quick turnaround via a merger with...