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I'm not seeing the erosion, but moving a lot of dirt around can get expensive, and nice retaining walls are even more expensive.
Are you wanting a patio behind that one tree for the shade? Or are you removing the tree? Kinda confusing.
But I can see adding vegetation and a path to a patio/structure. But is that where you want to sit, out by the road? It could be nice but I think you'd be adding lots of plantings.
The roof over the deck is not a huge job, but you have to make a solid structure and tie it into the house, cut the siding, add flashing, roofing, etc; by the looks of the existing deck it would be mostly rebuilt, too. If you did rebuild it maybe build it narrower (less depth from the house) that would make the roof look better.
Really, though, if shade is what you want put up nice awnings or a shade type of thing; I've seen some pretty cool ones that hang up like sails almost.
I had a neighbor years ago that was a really talented, I traded finish carpentry in his house for a professional design. It was worth it. So if you want to do something really nice, that's what I would recommend.
But on a frugality forum? Naw. Just get a big cafe umbrella in a heavy stand.
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We have a big umbrella up there, but unless we move it, the sun heats up the deck throughout the day. I’ve used it to cover me when grilling, but we have never eaten out there.
The patio would be under the deck. But it would be nice to have some plantings to provide shade and privacy.
We are planning to keep the tree. But would be ok removing it if it would help the overall plan. A friend had used a gazebo from Costco ($1500) on their deck for shade. I’m not sure if our HOA would allow it though.