The meaning of ‘scratch“ in "don't have time to scratch myself"
I know the sentence means too busy, but what does the "scratch" means literally here?

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It it was said by a woman, it refers to self satisfaction.

Answer

"Scratch" in this sentence means to drag one's finger nail over their skin. People scratch their skin where it "itches."
Scratching takes a few seconds, so the speaker is saying, "I am too busy to waste even a few seconds."