How to quote passage with literal error that changes the meaning?

If the purpose of quoting the text is to discuss the error, then quote the error. However, in this case, it seems that your purpose is to discuss the real situation. Quoting the error just makes things confusing. Instead, I suggest something like:

Wallace et al. [1, p.497] say that polynomial approaches, such as dispatching, "do not work well."¹

¹ [1, p.497] erroneously described dispatching as nonpolynomial; compare [1,p.495].

The footnote explains that the literal quote from the source is incorrect but attempts to downplay this. The point of the footnote is only to stop the reader being confused if they look up the quote, not to criticize the authors for the mistake. You might be able to phrase the footnote better than I have, to make this point more effectively.

But be sure that it really is an error. The phrasing "Nonpolynomial approaches, like dispatching, may not work well" is ambiguous and could mean any of the following things:

• "Nonpolynomial approaches, such as dispatching, may not work well." (That is, nonpolynomial approaches may not work well; dispatching is an example of a nonpolynomial approach. This is the interpretation that you've used.)

• "Nonpolynomial approaches that resemble dispatching may not work well" (That is, among the nonpolynomial approaches, those that resemble dispatching may not work well; other nonpoly approaches might be fine.)

• "Nonpolynomial approaches, which resemble dispatching, may not work well." (That is, nonpoly approaches resemble dispatching; nonpoly approaches may not work well.)

• "Like dispatching, nonpolynomial approaches may not work well." (That is, dispatching may not work well. Nonpoly approaches have the following commonality with dispatching: they also may not work well."

Also, as dan1111 points out in his answer, you need to be sure that your correction (changing "nonpolynomial" to "polynomial") fixes the error.

Avoid referencing this sentence altogether. It is just too confusing. Whether you quote it directly or not, someone who looks at the source will not find support for what you say, but will have to read and try to figure out what it is supposed to mean. This means that the reference is not serving its purpose of providing support for your statement.

• If the source has other text clearly making this claim, reference the other text. This is the content that actually supports the claim you want to make.

• If not, look for another source entirely. I would not base anything on a single sentence that appears to have an error like this. Because: how can you or the readers be sure of the intended meaning?

  • As David Richerby suggests, perhaps it was a poorly worded sentence, and another meaning was intended grammatically.
  • If there is an outright error, how do you know which word is wrong? You assume "polynomial approaches, like dispatching" was intended, but couldn't the intended meaning equally be "nonpolynomial approaches like X"? Where X is a nonpolynomial approach?
  • Even if you are sure, your readers might not be convinced. Questions about whether what you say is supported undermine the strength of your work.

Note: if the error has been corrected, as it turned out to be in this case, then it's clearly ok to reference the corrected version.

You should consider the possibility that you are misunderstanding the text.

It's possible that the author doesn't mean "Nonpolynomial approaches, of which dispatching is an example, ..." but rather he means "Nonpolynomial approaches, just as in the case of some polynomial approaches, ..."

In other words, his "like dispatching" doesn't mean that dispatching is an example of a nonpolynomial approach, but rather that it too suffers from the same problems as nonpolynomial approaches.

Simpler example: "Boys, like redheaded girls, have faces". This doesn't mean that redheaded girls are examples of boys. It means that boys, along with redheaded girls, share this property (having faces).

Just saying.

The author could clarify which is meant by using either "just as" or "such as" instead of "like".