My 1956 Mercury sounds like an old tractor, so I've been chasing exhaust leaks.  Driver side exhaust manifold was completely missing the gasket, plus half the manifold bolts were only finger-right.  I thought for sure that would solve my problem, but there's definitely another exhaust leak, right around this doohickey in the attached picture.  Looks like some type of heat riser?  Obviously not working, the lever just flops back and forth and it's not hooked up to anything.  Looks like a nail shoved in there at the top, and also looks like it might be cracked or broken around the manifold.

Can someone explain what this is and how it's supposed to work?

.It's a Heat Riser Valve. When its in complete Condition there is a Thermostatic Spring which closes the Valve causing an Obstruction for Exhaust Gas. It forces some Exhaust to go through the Passage in the bottom of the Intake Manifold and exits on the other side.Heats up the Intake Manifold to warm the Carburetor in cold Weather. As the Spring heats up the Valve opens allowing full Exhaust to flow through the Exhaust Pipe. You can replace it with either a new One or replace it with a Spacer available from most . Sellers like Macs or Tee-Bird. There is a Gasket between Spacer and Exhaust Manifold and between Pipe and Spacer. Thats probably burned out on yours causing the Leak.  The Heat Riser Valve is not really needed in warm Weather driving. I can't open your Picture for some reason. You can post Picture directly by clicking "Insert" at bottom of Post and then clicking the "Mountain with Sun" and download your Picture by selecting the Picture You want and click "open".

  

Thanks.  Here's another try at the picture:

It's a Heat Riser for sure. You might also need to pickup Exhaust Manifold Gaskets that go between Exhaust Manifolds and Head. Not originally used on these Engines but the Manifold Flanges can become uneven allowing Leak between Head and Manifold. I had to use them on Mine to fix Exhaust Leak  for that Reason.

  

I just replaced mine with a spacer I bought from NPD.  Slipped right in with no problem. I believe it's there to help the engine warm up.  Here in Florida we don't need any more heat.