This is a two-part fix, as you have a two-part problem. I think Jim is right, you've developed a path for the water build up that is flowing under your deck. Which means it's probably getting to the gunite and eroding dirt from there, too, which is pretty bad.
Pull the rock away from the deck a foot or two. Clean out any loose soil from under the deck. Pack concrete under the deck, as far back as you can get. That will do two things: fill in the gap some, and create a blocking wall that will help some with water getting under there, and it will help support the deck. Don't use dirt, don't use dirt with grass. The grass will eventually decompose and just leave loose soil, but either way the soil will just erode again. Concrete will be the better fix.
But that's not going to solve the problem from coming back. The erosion will just take up again a few feet away. To solve for that you need to reroute the water. So in that 2' gap you created in the rock, you dig a trench. Probably a foot deep. Looking at that last pic, it looks like the yard slopes a bit from right to left. Dig the trench that way. All the way past the pool. Hopefully you can get past that one palm tree in the way. And you extend the trench until the end is lower than where you started. Preferably somewhere that water can find a way to the street or some other drainage system. You lay in corrugated drainage pipe and that pipe will connect up two or three drain basins plotted in that wet corner. The height of the tops of the drainage basins is critical. Low enough to catch the water, but high enough that they don't fill with dirt. At the other end, if you can't connect the pipe to an existing drain system, you can use another drain basin. As long as the top of the exit drain is lower than the tops of the entry drains, the water will find its way away from that wet corner, even if water stays in the pipe. It'll do this:
Check out this page for ideas.
There are several types of drainage basins, here's one:
And several types of caps, here's one:
Back fill the trench, restore the gravel. If you don't build this drainage system, you'll never solve for the erosion.
Lowes or HD will have a selection of parts from which you can build what will work in your yard...
PS. Watering that lawn is a big part of the problem. You're probably getting runoff from the lawn year round, from rain in the winter and from irrigation during the summer. One of the drainage basins should be very close to where the lawn stops. Or do what I did, and replace the lawn with ground cover, watered with drip irrigation. Let's face it, the days of lawn in CA are over. Your deck is being ruined by the sprinklers same as mine was, too. The constant water will eventually create pits in the cream of the deck's concrete, if it's not happening already. I used Myoporum, which is great because its evergreen, never needs mowing (yay!) and grows from a central hub, which means you only have to water the center with drip, not the whole thing with sprinklers. It grows in about 9' circles. So the grass that's showing in that last pic would only need four or five Myoporum plants. Convert the sprinklers to a drip line with five emitters, bobs-yer-uncle. If you tear out the lawn, you can plot drainage basins not only in that wet corner, but all the way around the pool. It'll be a lot of work, but you'll be thankful you did it in a year or two. Just think of all the time and $ you'll save by getting rid of the grass. And no more grass clippings in the pool! It'll still look just as green, just as nice:
Dirk for such a well written and thought out reply!
