No, the fear is of the "getting hurt" part.
They don't fear the child. (If they did, it would be "a child who gets hurt.")
This is why gerunds (like "getting hurt") often take the possessive, so you can clearly see that it's acting as a noun.
I'm afraid of your getting hurt. -- The action of you becoming hurt is what I fear.
This is not the same as "I'm afraid of you getting hurt." -- I don't fear you. I fear the injury.
However, the possessive doesn't "sound right" when you say "afraid of a "child's getting hurt" so no one is likely to say or write it that way.
You must think of "a child getting hurt" as the noun phrase here.
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