Supposedly- according to what is generally assumed or believed (often used to indicate that the speaker doubts the truth of the statement)
Arrogantly- in a manner resulting from or revealing an exaggerated sense of one's own importance or abilities.
Eloquently- in a fluent or persuasive manner
Playfully- in a light-hearted manner
Blatantly- in an open and unashamed manner
The sentence talks about the kids who wanted to find their friend in a game. The blank requires an adverb to qualify the verb 'running' (to describe the way how the kids run). Options (a), (b), (c) and (e) 'supposedly', 'arrogantly', 'eloquently' and 'honourably' do not match the context of the sentence. The sentence talks about a fun activity and all the mentioned adverbs are either irrelevant to the context or not as appropriate as required and thus all the options are negated. Option (d) 'playfully' is the most appropriate adverb that matches the context of the sentence and so it can fill the given blank.
Hence, the correct answer is option (d).
Note:
Qualifiers: A word or phrase that limits the meaning of another word or phrase or makes it less general. 'Adverbs' and 'adjectives' are qualifiers.
Adverbs: Words that describe a verb, an adjective, another adverb, or even a sentence as a whole are called Adverbs.