Compacted clay soil says lack of organic matter. Core aeration is a short term fix that will need to be done annually, if not more often, if organic matter is not added to that soil.

You're doing fine, but I have some thoughts. Wood chips on top of the soil is no problem. That is where they belong. Wood must have fresh air to decompose. If you bury wood, it can remain wood for decades. Once you expose it to the air, it can rot away in a few months. And yes, when it is buried, the fungi that decompose it tend to 'rob' nitrogen from all sources. It just need the nitrogen from the air to get going.

You can soften your soil very easily. You need some clear shampoo, a hose, and a hose-end sprayer like the Ortho with the orange/yellow top. The application rate is a minimum of 3 ounces per 1,000 square feet. Maximum rate has not been determined but 50 ounces per 1,000 square feet seems to do no harm - just expensive. Start with 3. If your front yard is 1,000 square feet, put 3 ounces of clear shampoo into the bottle and add some water, milk, or molasses just to fill up the bottle. Then spray the entire contents onto the yard trying to get it all evenly. The dial setting doesn't matter although the high setting tends to empty the bottle so fast you can't get it on evenly. After you apply the soap, then water the lawn until you have about an inch of water on it. If you get runoff before you get an inch, then stop for 20 minutes while the soap and water soak in. Then resume until you have a full inch over the entire lawn. You can time your sprinkler by putting tuna or cat food cans out. Time how long it takes to fill them. Ranges I've seen are anywhere from 20 minutes to 8 hours. It depends on your sprinkler, water pressure, and hose. That deep watering is what you are supposed to be doing.

Where are you in the LA area? One inch per week is too much if you live in El Segundo.

Would you like to have a full St Augustine lawn? I can help with that. With subtle differences in watering and mowing you can get rid of bermuda. Bermuda in flower beds is a different story.

How high/low do you mow?
Have you had a lot of rain?

Can you post a picture of the yellowish grass? Preferably two pictures - both of them taken on a cloudy morning, one picture showing the entire lawn and one very close up showing the yellow blades. You might have iron issues or you might have a disease.

EB Stone's lawn fert is kind of bottom of the line organic stuff. It's not much more than feather meal. Feathers take decades to decompose and release the nitrogen within. What you get is a lot of fluff and just enough blood meal mixed in to give the grass a quick jolt. If you want to spend a lot less and get better results, find your nearest feed store and get a bag of alfalfa pellets (rabbit chow). Cost for 50 pounds should be on the order of $12.50. That seems to be a nationwide price for alfalfa pellets. Don't know why because there are a lot of factors going into pricing, but that's what everyone reports. The application rate is 20 pounds per 1,000 square feet. When you apply pellets, you must moisten them so they become soft. All that takes is a light spray, not a drenching. Moisten them and the next day they will have swollen and look like green worms. The alfalfa will fall into the grass by gravity or you can help it along by dragging a hose across them. Blood meal can work in a day or two to turn the grass green. Alfalfa and all the grain type fertilizers take three weeks. But the effect is longer lasting and better for the soil, in my opinion.

So, Dave, how does hugelkultur work?

Here is a link that might be useful: hugelkultur

Quoting from that link...

Another thing to keep in mind is that wood is high in carbon and will consume nitrogen to do the compost thing. This could lock up the nitrogen and take it away from your growies. But well rotted wood doesn't do this so much. If the wood is far enough along, it may have already taken in sooooo much nitrogen, that it is now putting it out!

I have a question regarding hard, compacted soil. We had a large hole dug to accommodate a 15-20 gallon mango tree. I ran some water in the hole last night and it is still there this morning--although not as much. Initially there were 3 large queen palms that I had removed with stumps ground out. There are still a lot of small roots from the palm trees. How can I get this water to drain, and is there anything than can be done to the soil so that it is not so compact.

Use it as an opportunity to introduce the shampoo at depth. Dilute a bottle of the baby shampoo into a five gallon bucket filled with water. Dump it into the hole and fill it again. You might also try to locate some soluble humic acid and add a pound or two dissolved first in the bucket. Only one shot so try to make the best of it to improve compaction. It is improve, not cure.