How can I edit the following texture...
...so it looks like this?
And how to create the burned spots
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For cycles. Use the texture to control how two different shaders mix:
(the color ramp is just to control the the intensity of the mix, but curves or a color bright/contrast node would work as well)
A different option is to use the image to control the color on a single shader:
Or use the image to control the mix and create some displacement.
Or additionally use it as a bump map:
I didn't have a way to make a normal map on this computer, but that would finalize the effect, with some good lighting effects like your reference image.