Since there's no context and there are several meanings of 「世話」, I guess this sentence means "that child is problem-free" or "that child is not being taken care of"
I don't think "that child is not being taken care of" is an option, but yes, the two options you listed based on the entry for 世話がない are both possible depending on context. It seems like the "problem-free" meaning often gets used in て form with すむ, so 世話がなくてすむ, but it seems like that's not always the case. So, without more info, I don't think it's possible to say for sure.
In my understanding only the second meaning would be appropriate here as in "The kid is hopeless / beyond help", with no statement about whether is actually trying to take care of it, just that there's no point to it.
加拿28大预测网
「あの子は本当に世話がない」？
Does it mean "that child is problem-free" or "that child is a real pain in the neck" ?