Oh, please help me in translating this sentence: "I’ve made a dog’s dinner of that. Let me just do it again." I understood that someone cooked a food for dog, and he says that he wants to do it again. Am I right?

If you interpret the sentence very literally, then yes, you would be correct. However, most likely, the sentence actually means something like "I've made a big mess of that, let me try that again." Also, as one might suspect, it's not limited to situations where food preparation is involved.

So, why is the phrase a "dog's dinner" is used to represent a general mistake or screw-up? I guess if one thinks of a dog's dinner as something sloppy, messy, or of low quality - that is, something that would be okay to serve to a dog, but not to a human - then it might make sense.