How can I fill the gap between layers of a stair tread?
Here's what I'd do to avoid hacky patch solutions that will probably fail:


Carefully remove the cove molding from under the original nosing. 
Cut the original nosing and the overlay flush with the riser to create a flat vertical plane. Use a circular saw, jigsaw, and a belt sander, if available.
Screw and glue down the overlay if it isn't already, so it's solid. 
Glue and screw a replacement nosing in place, matching the original size and profile, flush on top.
Reinstall the cove molding under the new nosing.
Fill fastener holes, sand, prime, paint.


This will leave you with a result indistinguishable from the original and just as solid.