Then why is poverty concentrated geographically in inner-cities and rural areas instead of evenly spread out? Obviously location and environment affect opportunity, including the perception that breaking out of poverty is even possible to begin with.
Because breaking out of poverty is a hard thing to do. It is certainly possible, beginning with a focus on achieving at a high level early in school (which can be a difficult task when dealing with life issues imposed by poverty). Doing extremely well in high school, which can lead to college, scholarships, etc. is one of the few exits on the freeway to adult poverty for impoverished children. Once that ship has sailed, they could go into trades, such as electrical and plumbing/heating. Most, won't, instead bouncing from job to job with no hope of ever improving their station in life, and will have their own children, instantly worsening their own financial situation and continuing the cycle.
But none of that means it's impossible or that it requires luck to break out of (again, assuming they live in a country with abundant economic opportunity). It requires hard work, discipline, and perhaps most importantly, an understanding that the limits their parents ran into that resulted in their impoverished upbringing are not insurmountable.
I would disagree with your luck statement. I broke out of poverty. My parents were both unskilled labor, we were on assistance at least once that I can recall.
Part of what allowed me to escape is luck. Yes it required that I developed early and so was ahead of my peers in education. It also required hard work to capitalize on that lead. However, what kept more than one of my cohorts from escaping was bad luck. A family member had a health issue, a legal issue, whatever. Their need to care for their family caused them to miss an opportunity (Get a job rather than go to college).
Unless you're advocating living life as a sociopath and ditching people as 'dead weight', luck is absolutely part of escaping poverty.
In my original comment, I said that there's no reason to be poor, with a couple of exceptions - including bad luck. With hard work and discipline, you don't need to have especially good luck to break out of poverty, but yes, any bad luck can wreak havoc on the life of anyone in almost any socioeconomic circumstance. This is why it's still important to have social safety nets, even if the general public doesn't like the fact that these nets will save both the very unlucky and irresponsible people/those that simply don't want to contribute to society.
Luck is luck. Whether you call it bad luck that they had an adversity to overcome that I did not, or good luck that I did not have that adversity, you're still talking about fortune.
In any case, whether you say it is that the parents are poor or that they are drug addicts or don't care or whatever, you're talking about things outside the control of the individual. I cannot choose the parents I was born to.