If by ‘abstract’, you mean ‘non-spatiotemporal’, then the answer is: no—such objects do not have causal powers.
If by ‘abstract’, you mean ‘such that they are themselves correlates of abstract conceptions’, then the answer is: sometimes—-it depends on the case.
Let me explain my meaning.
Consider the property of round or the property of being an even number. Neither of these objects exists in the spatiotemporal world, and neither, consequently, has causal properties.
(You may doubt their existence, and we can talk about that; but there is no denying that if they exist, then, being non-spatiotemporal, they cannot possibly affect or be affected by anything, since causal relations require locations and space and time.)
But there is another sense in which objects can be ‘abstract.’ The Ford Motor Company is, at least in some contexts, an ‘abstract entity.’ This entity is abstract in the sense that it would not even exist were it not for abstraction-heavy understandings among various people—-understandings relating to law, commerce, and the like. That said, the Ford Motor company most definitely does have causal properties.
(To be sure, whatever causal properties the Ford Motor Company has are mediated by the causal properties of non-abstract, lower level objects; but that is true of all objects having causal properties, except for the most micro-physical of micro-physical entities.)
The United States government is ‘abstract’, in the second sense of the term, and it has causal properties. The number 5 is ‘abstract’ in the first sense of the term, and it lacks causal properties.