This is a great example of the need to test driving the design to a good place. Ok, "good" is subjective but let's see:
That's going to be very hard to mock because you're hard coding a library class - maybe a file reader - in there.
Now I have control in MyApplicationClass and maybe I could have a testing switch to tell it to throw an exception.
Now I can plug in a mock helper that throws exceptions just for this test. My class under test no longer has a dependency on any class, just an interface. Reducing dependencies is a "good" thing, adding configurable flexibility might be a useful thing, too.
Another approach might be to move all the complex catch logic you want to do to another class and test it independently.
Now helper has a public method I can test.
Hope some of that rang some bells and looks useful.