I would make the trench no shallower than a foot deep. Plants should be fine with this even if planted on top and even if it hasn't been warm enough to break down before you plant. A foot deep is also enough to keep rain or snow or other weather from exposing any of it. You could mound the soil a bit over the top to keep it from sinking down and making a hollow.
If you decide to put it shallower than a foot deep, I would recommend digging it into the soil with a fork before you cover.
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