Adding support to existing retaining wall

The wall has a 4.5" concrete footing (as can be seen in the picture). Otherwise, I don't know how it's constructed. I was wondering about digging underneath the "gap area" and somehow filling down with concrete. Any thoughts on that?

Thanks for the advice! It sounds like having that gap for the BBQ is complicated and can affect the integrity of the wall. I am thus starting to think of having the wall continuous and somehow placing the BBQ with its profane tank above it. Any ideas there?

For the wall itself, the thought is to go with a dry stack stone retaining wall. Any suggestions on the structure of the wall and in particular on its drainage?

There's other options for placing the BBQ in the yard, but that area kinda fit well with a plan for a patio in that are. Also, the yard is not too big, so pushing the BBQ against the fence feels natural in terms of utilizing the space.

Yes, there use to be an old wood fence in front of that wall that covered the vinyl lattice. Once that old wood fence was removed, that ugly wall on the right was reveled.

For the wall structure, did you mean installing a perforated pipe? Could I get by with gravel only? The rocks will have gaps assuming it's dry stack wall, so I would think the water can find its way throughout the wall. Also, in general, I think the rocks are laid against the tilted soil in a dry stack wall so to use gravitation for holding the rocks in place. I am just not sure how that plays out with having gravel behind the wall.