In a way that is affectionate, tender, or characterized by fondling; with gentle caresses.
From 'fondling' (present participle of 'fondle') plus the adverb-forming suffix '-ly'. This double-suffix construction (verb + -ing + -ly) is common for creating adverbs from progressive verb forms.
English adverbs from '-ing' forms like 'fondlingly,' 'beguilingly,' and 'charmingly' have a special literary quality that makes them feel poetic—probably because they stack two suffixes, which gives them extra phonetic weight and formality.
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