There's a lot of different ways to tackle this problem, obviously, so I'll just point you in one of many directions. You'd have to consult the specific thinker's works to get their exact takes.
Okay, so the concept of 'human nature' in general needs some more definition, like you've said. Here's maybe a better phrasing of the problem: "Do humans have some sort of _innate_ properties that are shared amongst all humans?" This phrasing calls back to an old debate in philosophy: essentialism. This is a question of ontology, in other words, what kinds of things are there in the world, and how do we categorize them?
An essentialist takes the position that we can form categories of things by defining some set of properties, and then any entity which has those properties is one of those things. In other words, let's say something like "You're a human as long as you have a name." This is a position which is essentialist. And that might be a workable definition.
A non-essentialist may come along and challenge your ontology, though. "Ships also have names, but they are clearly not human." You may agree with this, and so you need to revise your set of properties. "Humans are things that have names, and walk on two legs." The non-essentialist may respond, "birds have names, and when they walk, they use two legs. Your ontology is incorrect.
You see where this is going. Those who take a position against essentialism claim that this approach to categorization is a folly. It's fundamentally broken.. To give you a counterpoint, I generally agree with a process philosophy, which means that I equate identity with _doing_, not with _being_. An example of a process philosopher's standpoint on ontology: "You're a Rubyist if you write Ruby code." This kind of categorization is more useful, because over time, my identity can change. I might be a Rubyist for a while, but at some point, I wasn't. And I might stop being one in the future, too. An approached based on do-ing includes things like a time component that an essentialist position never can.
So, returning to the original question, no, the question of an inherent nature doesn't make any sense, and therefore, neither does 'human nature'. So where do we go from there? Well, any claim that's reliant on an appeal to human nature is incorrect. For example, "Humans are inherently greedy."
Anyway, that's a hastily-typed overview of a concept in a HN comment, but hopefully that helps you get the gist of it.