I'm not sure I remember this the right way, but here's an attempt: 

"Constructivism" can refer to a family of normative-ethical views according to which objectively right moral facts are whatever would be the output of some constructive function, such as an imagined social contract or the Kantian realm of ends. "Constructivism" can also refer to a non-realist metaethical view that moral language doesn't refer to moral facts that exist in an outright objective sense, but are instead "construed" intersubjectively via some constructive function. 

So, a normative-ethical constructivist uses constructive functions to find the objectively right moral facts, while a metaethical constructivist uses constructive functions to explain why we talk as though there are some kind of moral facts at all, and what their nature is.

I'm really not sure I got this exactly right, but I am confident that in the context of this "letter to a young philosopher," the author meant to refer to the metaethical version of constructivism. It's mentioned right next to subjectivism, which is another non-realist metaethical position. Unlike some other Kantians, Korsgaard is not an objectivist moral realist. 

So, I think the author of this letter is criticizing consequentialist moral realism because there's a sense in which its recommendations are "too impartial." The most famous critique of this sort is the "Critique of Utilitarianism" by Bernard Williams. I quoted the most relevant passage here. One way to point to the intuitive force of this critique is as follows: If your moral theory gives the same recommendation whether or not you replace all existing humans with intelligent aliens, something seems (arguably) a bit weird. The "human nature element," as well as relevant differences between different people, are all lost! At least, to anyone who cares about something other than "The one objectively correct thing to care about," the objective morality will seem wrong and alienating. Non-objectivist morality has the feature that moral actions depend on "who's here." That morality arises from people rather than people being receptacles for it. 

I actually agree with this type of critique – I just wouldn't say that it's incompatible with EA. It's only incompatible with how many EAs (especially Oxford-educated ones) currently think about the foundations of ethics.

Importantly, it doesn't automatically follows from this critique of objectivist morality that a strong focus on (some type of) effectiveness is misguided, or that "inefficient" charities suddenly look a lot better. Not at all. Maybe it can happen that certain charities/projects look better from that vantage point, depending on the specifics and so on. But this would require further arguments.