Learn Arabic (Modern Standard): ركض الطفل مذعورا.

That extra alif holds the "an" ending (tanween fatha) for the accusative case. In this sentence, مذعورا is a "Hal" (a word describing the state of the subject during the action). In Arabic, words used as a Hal must always be in the accusative case.

Since الطفل has "al-" (the), why doesn't مذعورا also have it?

Because مذعورا is not an ordinary adjective here. If it were, it would match the definiteness and mean "the terrified child ran." Instead, it is an Adverb of State (Hal) describing how the child ran. The strict rule for this structure is that the subject must be definite (with "al-") and the Hal must be indefinite (without "al-").

How would the sentence change if "the child" was a girl?

You would need to make the verb, the subject, and the Hal all feminine. "The girl" is الطفلة, so the verb becomes ركضت. The Hal must match the subject's gender, becoming مذعورة (ending in a taa marbuta). The full sentence would be ركضت الطفلة مذعورة. Notice that the taa marbuta takes the accusative "an" sound but does not get an extra alif added to it.

Is مذعورا an adjective or an adverb?

In terms of its dictionary form, it is an adjective (specifically, a passive participle meaning "terrified"). However, because of how it is used in this sentence (indefinite and accusative), it functions as an adverb of state. Arabic commonly uses adjectives in this specific way to show the condition someone was in while performing an action.