Non-marxist "leftist" objections to Marxism (libertarian socialist/synthesis anarchist) here. My main “issues" with Marxism: I don't think the historiology, what is called “dialectal materialism” or “historical materialism”, is universally applicable. I think it is applicable as an analytic or descriptive theory in limited contexts.
I am skeptical of most “progressive" accounts of history and historical meta-narratives in general. In that sense I am a Foucauldian.

(Charles Brown: Foucault has a fool's cult)

I don’t think marxism has a sufficient theory of how the concentration of political and social power is as much a source of as it is a product of class-stratification. This in turn leads to the bureaucratic classes arising in ostensibly “socialist” countries.

There are certain political issues that marxism (due to it being primarily a theory of class) really doesn’t address due to its emphasis on being “materialist.”

I think the idea that marxism is a science (in the same way as say electrodynamics or genetics) is wrong. Marxism has very limited and variable predictive power, in my opinion.