Unit testing with mockito for constructors

You can use PowerMockito

See the example:

Second second = Mockito.mock(Second.class);
whenNew(Second.class).withNoArguments().thenReturn(second);

But re-factoring is better decision.

Here is the code to mock this functionality using PowerMockito API.

Second mockedSecond = PowerMockito.mock(Second.class);
PowerMockito.whenNew(Second.class).withNoArguments().thenReturn(mockedSecond);

You need to use Powermockito runner and need to add required test classes (comma separated ) which are required to be mocked by powermock API .

@RunWith(PowerMockRunner.class)
@PrepareForTest({First.class,Second.class})
class TestClassName{
    // your testing code
}

Once again the problem with unit-testing comes from manually creating objects using new operator. Consider passing already created Second instead:

class First {

  private Second second;

  public First(int num, Second second) {
    this.second = second;
    this.num = num;
  }

  // some other methods...
}

I know this might mean major rewrite of your API, but there is no other way. Also this class doesn't have any sense:

Mockito.when(new Second(any(String.class).thenReturn(null)));

First of all Mockito can only mock methods, not constructors. Secondly, even if you could mock constructor, you are mocking constructor of just created object and never really doing anything with that object.