In a manner of making soft cooing sounds, or speaking in a soft, affectionate, murmuring way.
From 'coo' (to make soft sounds) plus the adverbial suffix '-ingly,' which converts adjectives and verbs into adverbs describing manner. This is a standard English word formation pattern.
The suffix '-ingly' is incredibly productive in English, meaning speakers can add it to almost any verb to create an adverb—you could theoretically say 'he spoke runningily' and people would understand you, even if it's not a real word yet!
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