A few years ago I wrote a long essay arguing against the idea that poverty or inequality causes crime: https://devinhelton.com/inequality-crime
The key argument to me is that we have too many examples of extremely poor, unequal, and or marginalized societies with very, very low crime -- crime much lower than that of the modern U.S. where even the poor are richer than those living in the slums of yesterday. Examples of such low-crime, poor areas: post-World War II South Korea, the slums of Edwardian England, Chinatowns in 1900s America, etc. Nor are economic depressions all that correlated with rises in crime.
Conversely we also have examples of extremely high rates of crime among the upper class in some societies, such as feuding clans during the late Middle Ages or wild college students in certain university towns. (Although usually when nobles commit crime against each other or their serfs we call it "war" or "oppression" or "predation" and not "a high crime rate.")
It seems to me that crime primarily flows from a breakdown in the social fabric. Crime flows from a breakdown in virtue, a breakdown in institutions that enforce the law, a breakdown from a cooperate-cooperate equilibrium to defect-defect.
The correlation between poverty and crime probably runs the other direction -- the anti-social tendencies that make a person a criminal also make it hard to hold down a high-paying job. Communities that have a very crime rate make it very hard to support and sustain productive businesses.
Why do so many think that it is poverty that causes the crime?
I think it is political. It is people to the left-of-the-center that really push the idea that poverty causes crime and I think they are very loathe to admit that the old-school conservatives were right about things like policing mattering and culture mattering for the prevention of crime. Also, politicians like to frame problems in ways that make their voters look like the victims, not the source of the problem. It is the Democrats who rely on higher-crime demographics as their voting base, so Democrat politicians don't want to be "victim blaming" their own voters as being the real source of the high crime.
Furthermore, there is a lot of tricks played with the definition of poverty, with the more recent definitions expanding the concept into basically including all sorts of bad things, which then means there are an infinite number of programs that can be created and infinite number of jobs created in order to fight poverty. Rather than just give people money via something like Nixon's negative income tax, you can create all sorts of jobs and power for left-of-center institutions and bureaucracies. Here are the modern definitions of poverty:
United Nations Definition of Poverty: “Fundamentally, poverty is a denial of choices and opportunities, a violation of human dignity. It means lack of basic capacity to participate effectively in society. It means not having enough to feed and cloth[e] a family, not having a school or clinic to go to, not having the land on which to grow one’s food or a job to earn one’s living, not having access to credit. It means insecurity, powerlessness and exclusion of individuals, households and communities. It means susceptibility to violence, and it often implies living on marginal or fragile environments, without access to clean water or sanitation.” https://www.learningforjustice.org/sites/default/files/tt_poverty_h1.pdf
World Bank: Poverty is pronounced deprivation in well-being, and comprises many dimensions. It includes low incomes and the inability to acquire the basic goods and services necessary for survival with dignity. Poverty also encompasses low levels of health and education, poor access to clean water and sanitation, inadequate physical security, lack of voice, and insufficient capacity and opportunity to better one's life
Two things to notice about these definitions.
First, notice that with the inclusion of "susceptibility to violence" in this definition, the link between poverty and crime within a community becomes a tautology.
Second, there is so much gathered under this definition, that it gives every powerful left-of-center institution something to do in order to fight poverty and thus, in their argument, to fight crime. To fight crime, fight poverty. To fight poverty, fund the schools, fund the universities, fund healthcare, fund social workers, fund public housing, fund doctors, etc. etc. Fund all the institutions that make up the base. No need to fund the police, though. Better to shift funding from the police to all these other things in order fight the root causes of crime. The argument that poverty causes crime thus leads to funding the institutions of the left-of-center faction rather than institutions of the right-of-center (churches and police).