In a kind, compassionate, and sympathetic manner; with gentle and benevolent intention or feeling.
From gentlehearted (adjective) plus the adverb-forming suffix -ly. This adverb allows describing actions performed with kindness and good intention.
Adding '-ly' to adjectives transforms them into adverbs describing how something is done, but notice how rarely we use 'gentleheartedly'—most people just say 'kindly.' This shows how some grammatical forms survive in the dictionary even when speakers stop using them.
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