In a way that expresses earnest pleading or desperate request; with urgent supplication.
From 'beseeching' (adjective/participle) plus the adverb suffix '-ly,' which converts adjectives and participles into adverbs describing how an action is performed or a state exists.
When a character in a novel 'looked beseechingly' at another person, you instantly feel the emotional weight—it's a single word that conveys complete vulnerability. The '-ly' transforms the image into an action, making the desperation active and immediate.
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