Abstractions are mind-dependent in the sense that they do not interact with the physical world except through the medium of the mind. Abstract objects have no mass and exert no forces; they are causally inert, yet they can influence physical events thorough the medium of mind. For example, I might go down a line of traffic cones knocking over every third one because I'm superstitious about groups of three. There is a sense in which the number three has influenced the world.
Physical objects are mind-independent in the sense that they interact with the physical world without the medium of the mind. However, there is an important sense in which physical objects are constructed by the mind as Armand suggested in his comment. There are obvious examples such as a place setting consisting of knife, fork, spoon, plate and glass. The only thing that makes this collection of unlike things into a single object is the way that the mind views them. If you follow this sort of logic to its core, you eventually have to conclude that many (and perhaps all) physical objects are essentially just objects because that is how they are perceived. However, despite being, in a sense, mentally constructed, physical object still engage in physical interactions independently of the mind. This is how they are different from abstractions.
So, in answer to your questions: no, there aren't any physical objects that are abstractions because physical objects are mind-independent (in the sense I described) and abstractions are mind-dependent. And, yes, this is one of the many reasons that physical reductionism fails: because abstract objects cannot be reduced to physical objects.
In the comments, @causative notes that a computer program could also do something like knock down every third traffic cone and asks how this is different from a human mind doing it.
My response is that the computer is a physical device that operates on physical causality, so the proper way to evaluate the process is with physical theory. In physical theory, causes are forces, and the number three exerts no forces, so it cannot be a physical cause.
By contrast, my story about knocking over traffic cones was about mental processes. When speaking about mental processes, you use the language of belief, intention, thoughts, etc. It is a different language and requires a different theory of causation. Within this language of mental processes, the number three had an effect.
Now, this dichotomy of language is not by itself controversial. There is no serious thinker who would deny that, in mental language, the cause of me knocking over the traffic cones was my fear of the number three. What is controversial is the connection between my intention to knock over the traffic cone and the physical action of my body to do so. Causative seems to want to deny me the use of mental language on the grounds that I don't have a reduction of that language to the physical, but since no one has a reduction of mental language to physical, that is an unreasonable demand.
Also in the comments, J.D. implies that my use of two different languages means that I am a Cartesian dualist, but one doesn't have to be a dualist to recognize that there are two different languages of causality and that there is no existing reduction of one to the other.