Very much; in a way that causes great pain, regret, or need; seriously or greatly.
From Old English 'sar' (painful, sore) plus the adverbial suffix '-ly.' Originally meant 'in a manner causing soreness or pain,' but evolved to mean 'greatly' or 'painfully' in an emotional sense.
English has this interesting pattern where words for physical pain become words for emotional intensity—'hurt feelings,' 'broken heart,' 'aching desire'—and 'sorely' is a perfect example of that metaphor in action, showing how our brains map emotional suffering onto bodily pain.
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