What was the relationship between slaves and poor Romans needing jobs in Rome's labour market?

Not much to go on historically.

Ancient Rome changed rules of citizenship over time and the rights of different classes. Patricians at the top, then different classes of Plebeians (think middle class with upper and lower), Proletarians (working poor who didn't own anything), Non-citizens but free, and Slaves.

Remember that slavery was generally different for most of the world and history than our idea of American slavery. That sort was reserved for criminals, be they citizen or slave, they ended up working mines or other undesirable jobs. All that you can read from many sources.

Competition was generally between the citizen classes with the Plebeians gaining more rights and power as time went on; gaining ground on the Patrician class (the rich) and increasing their representation in government.

Because they had a top-down voting system, the Proletariat (theoretically having the last vote but no direct representation) didn't get much say, if any at all. Easy to find this information.

There are many examples of how the dynamics between the lower classes play out in our own history and in modern society. The majority of Arab countries have populations without civil rights; non-citizens who have no legal recourse even if cheated financially or harmed physically. Mostly the same for undocumented workers in the USA. The poor everywhere usually have limited ability to vote and little direct representation.

Political power is (almost) always tied to wealth, and the Romans actually had tables for wealth vs citizenship status (https://en.wikipedia.org/wiki/Social_class_in_ancient_Rome); and were quite frank and upfront about it. Generally when there were lots of slaves from war, the poor weren't forced to do the worst work but if there weren't enough they would. Rome invented serfdom due to labour shortages; how they did this is by raising taxes on the poor and if they couldn't pay they had to work (farming and mining). On the other hand, not all slaves were poor and they were permitted to own property.

Rome like the British colonies and later like the US had to take care to not piss off all these groups at once. Whether poor proletarians, poor non-citizens, or poor slave were usually in the same boat and knew it.

In the American colonies the poor, indentured servants, actual slaves, and Native Americans banded together often enough to cause the wealthy problems. By granting different rights to different groups (black slaves had to carry papers, Natives had to leave town by dark, criminalizing intermarriage, etc) and hiring former soldiers and poor citizens as a police to enforce policies, the wealthy (in Roman the Patricians and upper-Plebeians) kept them divided - most of the time.

Interesting to note that in the Colonies as in Rome there was no racial divisions, discrimination, among the common people until they were legislated and enforced for generations.

"Gangs of New York" tells what happened when the slaves were freed; then the competition and discrimination started, but it was often initiated and fanned by financial and political interests to keep wages down (read any history of racial riots in US); and one group being used as a scapegoat for the other's lack of power and earnings.

In Ancient Rome, 6-12 (different times, and how you count) distinct classes were created with differing rights. Once again, they were upfront about it while in the US we don't formally acknowledge a class system though it's almost completely parallel to the Roman's. Just count prisoners, felons, and wage slaves as the slaves.

Howard Zinn's book People's History of the United States documents all of this and perhaps the most telling thing is how southern states had to formally outlaw slaves giving their own food to poor white families. Rome did not have TV so it wasn't as easy to divide the poor, get them to blame each other for financial inequality rather than look at those with the wealth, so I expect it wasn't an issue.