Does this study account for the obvious: poor people eat more UPFs and are more depressed?

Yes

The authors of the study accounted for income disparities:

with adjustment for known and suspected risk factors for depression, including age, total caloric intake, body mass index (BMI; calculated as weight in kilograms divided by height in meters squared), physical activity, smoking status, menopausal hormone therapy, total energy intake, alcohol, comorbidities (eg, diabetes, hypertension, dyslipidemia), median family income, social network levels, marital status, sleep duration, and pain.

https://jamanetwork.com/journals/jamanetworkopen/fullarticle/2809727