Blurry core of low-quality camera

I am doing some experiments to improve the image, so I am taking photos from my cheap camera. The camera has mosaic artifacts and all the images look like a grid. I think that the core of the bunkers (out of focus) and the Gaussian core would not be the best candidates. Any suggestions?

EDIT :
Example

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I suspect that this cannot be done with a permanent kernel, because the effects on the pixels do not match (therefore there are β€œmeshes”).

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