In the manner of begging; like a beggar would request or ask; in the style of someone seeking charity.
From 'begging' (Old Norse 'beggja') + '-wise' (Old English '-wisan'). A compound adverb combining action verb and manner suffix.
Medieval and Early Modern English loved '-wise' compounds to describe manner—'beggingwise' is now archaic, but it shows how productively English speakers created new adverbs.
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