The biggest problem with this study is they did not find what they hoped to find, or at least what would have been most interesting. They broke their subjects up into 4 groups based on early and late socioeconomic status, High/High, Low/Low, High/Low, Low/High. They only found differences between High/High and Low/Low. If there were some big differences differences between DNAm resulting from early poverty, I would expect the Low/High to have had significant differences. In other words they did not prove or disprove the click-bait title "Poverty leaves a mark on our genes".
Of course the lack of "significant" differences could be a power (sample size) issue as the number of upwardly (or downwardly) mobile people was half of what the maintainers in either class was.
I agree this is a clickbait article, but I do not think the research (in the paper) is bad. Hopefully this paper will be the basis of a higher powered grant to follow up on their findings.