With no argument:
ourObject = PowerMockito.spy(new OurClass());
when(ourObject , "ourPrivateMethodName").thenReturn("mocked result");
With String
argument:
ourObject = PowerMockito.spy(new OurClass());
when(ourObject, method(OurClass.class, "ourPrivateMethodName", String.class))
.withArguments(anyString()).thenReturn("mocked result");