Is "as that of drugs can do* or "as that of WHAT drugs can do" correct?

I think a better rephrase would be this:

The issue is that abuse of "Kratom" leaves can have the same deleterious effects as drug abuse.

"An abuse" is not idiomatic, because "abuse" is an ongoing thing.

"That of what drugs can do" is hard to parse, but because the writer in the comments indicated that he wanted to emphasize the parallelism of "abuse" it would be better to just include the word twice instead of writing around it with wordy clauses. In the rewritten sentence, the implied portion dropped off the end is "can have."

First, I'm not a fan of either sentence; however, no. 2 is correct. I would personally not say or write it so wordily; I would say or write it like this:

The issue is that an abuse of "kratom" leaves can have the same deleterious effects as drugs can have.