This post reproduces, in slightly amended form, two emails of mine that were part of an exchange about the meaning of 'human nature'.
1. I don't believe I've misunderstood the import of Oliver Burkemann's article. Pinker ascribes things to human nature, sees them as hardwired, and his critics think those things aren't hardwired. If this were all there was to it, I wouldn't have bothered to post.
But it's not all there is to it. As the views critical of Pinker are presented in the article, his critics say not merely that his version of human nature is wrong but that there's no such thing as a human nature. Hence, 'the belief that there is such a thing as human nature... has proved incendiary'; and the 'belief in a "blank slate"'; and 'how could change happen if people were born with their aptitudes and character traits hardwired?'; and 'That doesn't mean... such a nature actually exists.'
Would-be progressives often deny there's a human nature when they want to challenge a particular conception of what human nature is, of the traits human nature contains. The latter purpose is perfectly legitimate, whether the challenge is vindicated or not in the given case; but the denial that serves as its vehicle is a nonsense.
2. I think you're conflating the meaning of the term 'human nature' with different claims about what human nature is, different conceptions of it.
So, for example, if somebody says that it's futile to aim for a society from which all cruel and aggressive behaviour has been eliminated, because there are cruel and aggressive impulses ineradicably built in to our human make-up, there would be nothing absurd or nonsensical about denying this claim. Whether the claim is right or the denial of it is right I leave to one side, but it's a perfectly meaningful disagreement.
But if the person denying the claim says not that human nature isn't like that (cruel etc) because it's like this (benign, or open to being influenced in either cruel or non-cruel ways), but rather that there's no such thing as a human nature since human beings are born totally blank, as it were, and 'culture [in a phrase of Richard Rorty's] goes all the way down', then that aforesaid person will be embarrassed in having to deal with some other hypotheses: such as that there are human-natural impulses towards self-preservation and a degree of self-interest, that there's a susceptibility to pain, a capacity for conceptual thought of some complexity, and so forth.
You speak of what's 'generally' meant by 'human nature'. What's generally meant by it is some bundle of more or less enduring characteristics that are transcultural and transhistorical, all the variations due to language, culture and socialization notwithstanding. There has been wide disagreement about what is and what isn't part of human nature, but that is something else.
The suggestion that there is no human nature is preposterous, a bit like the claim would be that dogs don't have a nature because they can be trained up in different ways - which, of course, no one would ever say about dogs. Human beings are much more adaptable, much more capable of 'remaking' themselves, than dogs are - a possibility based, indeed, on their distinctively human capacities - but they remain biological beings with a natural endowment of relatively permanent attributes: abilities, needs and limits. Pinker may be wrong about what some of these natural attributes are but he's not wrong that there's a human nature.