Earlier quoted context omitted.
The cohorts are huge (120,000+ people) and they controlled for a number of other factors in the analysis, so I don't think sample error is here to blame for why they observed high-pesticide fruits/veggies having less health benefits than low-pesticide fruits/veggies.
The p values are small enough that the correlation seems likely to be real, even after lots of dredging through that dataset. I'm less sure about the causality, though. It's also possible they've just discovered that richer people (a) live longer, and (b) like organic produce. They controlled for various factors (note 1 on table 2), but none that seem like usefully strong proxies for income or wealth.
I'm not sure how controlling for income/wealth would matter, given that the study is all nurses and they control by ethnicity.