The Pleiades, also known as the Seven Sisters, are an open star cluster named after the seven divine sisters of ancient Greek mythology. Plato was a famous Greek philosopher, one of the greatest and most influential thinkers of all time.

Markham is referring here to the chronic lack of opportunity for the working man to do anything more than slave away each day for an absolute pittance. All he can do is toil away for his lord and master, gradually turning himself into a "monstrous thing distorted and soul-quenched".

As such, he has no time to contemplate the higher things in life, whether it's the wonders of the heavens or the great ideas of the past; they mean nothing to him. The speaker makes it abundantly clear that this is no kind of life, certainly not the one that man was originally created for. As well as work, there must be time for study and contemplation.

In this complex stanza, points are being made in reference to a Biblical allusion to Psalm 8:5: "You have made [humans] a little lower than the Angels and crowned [humans] with glory and honor" (Psalm 8:5). To describe laborers—as seen by him in the oil painting "L'homme à la houe" (1863) by Jean-François Millet—Edwin Markham enlarges the Biblical concept of "a little lower," making it a great "gulf" of separation between the laborer and the angels (seraphim). He assigns "Time's tragedy" to the laborer's "aching stoop." Dramatizing his point for the learned (non-laborers) who are reading his poem, he suggests that such stooped, lowly, "profaned and disinherited" laborers have no strength to consider the problems Plato poses or the theories he presents; they have no opportunity to unbend and turn their eyes skyward to contemplate the meaning of the movement of the starry Pleiades. 

What gulfs between him and the seraphim!
Slave of the wheel of labor, what to him
Are Plato and the swing of the Pleiades?

Markham goes on to suggest in the lines following "Pleiades" that the "dread shape" of the laborer, through whom "the suffering ages look," does not have the time, strength, or ability to contemplate the mundane, earthly pleasures of the "peaks of song, / The rift of dawn, the reddening of the rose." Markham makes the point that grueling labor reduces humans to being far lower than their God-given state, reduces them to a state that is deadened to higher thought and contemplation, to a state that is as stooped as their physical state:  

Time's tragedy is in that aching stoop;
Through this dread shape humanity betrayed,
Plundered, profaned and disinherited.

The larger point that these lines are part of, "what to him / Are Plato and the swing of the Pleiades?" is that "this dread shape humanity betrayed"—plundered of rightful thought, profaned of rightful place, disinherited of the the wonders of life—cries out in protest to the "Powers" that created life. His cries, which are silenced along with his contemplation of Plato and his wonder at how the summer and winter "swing" of the rising and setting of the Pleiades cluster, are a prophecy of retribution for those humans who lower other humans in this way: "Cries protest to the Powers that made the world, / A protest that is also prophecy."

These lines from Markham's "The Man with a Hoe" mean that working people have no spare energy or time for subjects that require intense thought and contemplation, such as the philosophical writings of Plato and the behavior of star clusters such as the Pleiades. Instead, the working person who Markham describes in his poem is "slave to the wheel of labor," meaning that working people are always working as if they were slaves and are never released from their labor. The working person also does not have time to enjoy delights such as songs, the sight of the dawn, and the way in which roses blossom. Instead, the working person is, like an ox, working so much that he or she has no time for contemplation of abstract thought or for the appreciation of nature.