Yes, it's an interesting question that has caused many people problems. As you have said, you have challenges with the , having clause.

Really, it's just a way to join two sentences. Here we have two sentences:

Mary has finished her homework.
Mary is going to the party.

We can combine these two sentences into: Mary, having finished her homework, is going to the party.

Here's a similar question:

Neuroscientists, having amassed a wealth of knowledge over the past twenty years about the brain and its development from birth to adulthood, are now drawing solid conclusions about how the human brain grows and how babies acquire language.

(A)Neuroscientists, having amassed a wealth of knowledge over the past twenty years about how the human brain and its development from birth to adulthood, are
(B)Neuroscientists, having amassed a wealth of knowledge about the brain and its development from birth to adulthood over the past twenty years, and are
(C)Neuroscientists amassing a wealth of knowledge about the brain and its development from birth to adulthood over the past twenty years, and are
(D)Neuroscientists have amassed a wealth of knowledge over past twenty years about the brain and its development from birth to adulthood,
(E) Neuroscientists have amassed, over the past twenty years, a wealth of knowledge about the brain and its development from birth to adulthood,
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The first sentence is really just two sentences combined:

Neuroscientists have amassed a wealth of knowledge over the past twenty years about the brain and its development from birth to adulthood.
Neuroscientists are now drawing solid conclusions about how the human brain grows and how babies acquire language.

P.S. Part of your problem may be that the first answer choice seems incorrectly typed. I believe that it should say "prescribed a drug at a dosage too low to be effective..."

Thank you very much for the explanation.
I can understand the examples you gave, but still struggling about the original problem. Can you help to explain the original problem? Thanks in advance.