Marx stressed the historical specificity of social practices and institutions. Did he think that human nature was similarly historically specific and aren't his views only of antiquarian interest in view of the findings of modern biology ?
▻ MARX AND CONTEMPORARY BIOLOGY
Marx knew nothing of the findings of present-day evolutionary psychology, neurophysiology, neurochemistry and ethology, and of the empirically-based information they provide about human nature. So much is obvious.
But we need to be aware of his normative stance towards human nature. He did not offer, or claim to put forward, a scientifically informed view - a view up to speed with the 21st century - of the nature of human beings. He would not have wanted to contradict scientific findings but he was not in competition with them. When he conceptualised human nature he did so from the standpoint of an emancipatory revolutionary. That at any rate is how he saw matters. It is not evident that 21st-century biology undermines his ideas about human nature.
'Innate' would not be my preferred term. Marx did not think in terms of a primordial, pre-evolutionarily fixed human nature. He could readily agree that, as we have evolved, this is the nature we have. It is not artificial or contingent on social conditions.
From his limited, or at any rate specific, perspective he had views about the conditions in which human beings could flourish given their 'species-being'. Okay, jargon : but not empty or pretentious jargon. I don't intend a defence of Marx, only a clarification.
▻ HUMAN NATURE - SPECIES-BEING
Marx's view is that we have 'species-being' (Gattungswesen). This is another way of saying that we have an innate human nature - that there is a human essence. He rejects the Christian view that work or labour is 'the curse of Adam' which ideally we would be well rid of. His view is also that under capitalism we are exploited and alienated workers at worst and aspirant consumers at best. I am only outliving my understanding of what I think he believes.
Marx is definite that we are active, creative, productive beings by nature; and that we can achieve our deepest satisfaction from developing and exercising skills, overriding obstacles, and otherwise structure the world. Work or labour does not simply produce a result that can be consumed; it transforms the labourer as action, creation, and production draw out his or her 'human freedom'. The world that is right for our species-being is one of freely-chosen creative activity.
▻ CONDITIONS FOR HUMAN FLOURISHING
That freedom is not complete until work or labour takes places under conditions quite different from those that prevail. What Marx has centrally against capitalism is not its exploitation (Ausbeutung) the worker (though he thinks it does exploit, through the extraction of surplus-value or Mehrwert, i.e. selling the products of labour above their cost of production without benefit to the worker) or that it induces alienation (Veräußerung), estrangement of the worker from what he or she produces (which is not his or her own). It's rather that capitalism treats the worker as someone to be kept alive, to reproduce further labour, and to consume. The capitalist form of social organisation has no aim or incentive to create the conditions to enable - to repeat the phrase - freely chosen creative activity.
This is (to repeat) simply an answer to the question, not an endorsement of Marx's views or a critique. In my view Marx did believe that there is an innate human nature; I have tried to spell out what he believed it was. His views were not science but it is not clear that, if empirical contemporary biology cannot verify or confirm his normative view of human nature, it does anything to contradict it.
K. Marx, 'Economic and Philosophic Manuscripts of 1844', tr. M. Miligan, NY : Dover, 2007. K. Marx & F. Engels, 'The German Ideology', 1845/6, London : Lawrence & Wishart, 2004. Norman Geras, 'Marx and Human Nature', London : Verso, 1983.