What the research actually found

Richardson T, Elliott P, Roberts R (2013). The relationship between personal unsecured debt and mental and physical health: a systematic review and meta-analysis. Clinical Psychology Review, 33(8), 1148 to 1162 doi

Debt is associated with depression, anxiety, problem drinking, drug dependence and suicide. The authors are explicit that the great majority of the studies looked at people at a single point in time, so the association is well established and the direction of cause is not.

Sweet E, Nandi A, Adam EK, McDade TW (2013). The high price of debt: household financial debt and its impact on mental and physical health. Social Science and Medicine, 91, 94 to 100 doi

Different kinds of household debt do not sit on people in the same way. Short term, high cost unsecured borrowing carried the heaviest load for mental and physical health, which is why treating all debt as one moral category is inaccurate as well as unkind.

Ridley M, Rao G, Schilbach F, Patel V (2020). Poverty, depression, and anxiety: causal evidence and mechanisms. Science, 370(6522), eaay0214 doi

Reviewing evidence that includes randomised interventions, the authors argue the relationship runs in both directions: poverty contributes causally to mental illness, and mental illness deepens poverty. They note that much of the causal evidence comes from low and middle income settings, so it does not transfer automatically everywhere.