This is an opinion piece that just came out in JAMA by David S. Ludwig, MD, PhD; Susan J. Blumenthal, MD, MPA; Walter C. Willett, MD, DrPH; I have excerpted it here, because it is behind a paywall, so I can't link directly to it. I hope this isn't illegal, but I think its an important document, so I'm posting it anyway.

Historically, the image of food insecurity in the United States was an underweight child with inadequate calorie intake. However, government food assistance programs including SNAP have been successful in reducing, although not entirely eliminating, this form of deprivation. Today, however, the picture of food insecurity is increasingly an overweight or obese child consuming a poor-quality diet.2 Indeed, the highest rates of obesity are found in people with the lowest incomes. Among poor populations, 7 times as many children are obese as are underweight.3 Typically, the challenge for low-income families in today's modern food environment is not obtaining enough food, but rather having dependable access to high-quality food. An estimated 16.7 million youth younger than 18 years do not consistently know when, or how adequate, their next meal will be.4

Studies in animals provide insights into the relationship between food insecurity and obesity. In one experiment, normal mice exposed to random, intermittent fasting developed increased body weight and fatness compared with animals with predictable food availability.5 Moreover, the intermittently fasted animals had increased expression of a hunger-stimulating brain neuropeptide and decreased body temperature, biological changes that would predispose to weight gain. Although these studies were conducted in animals, parallels might be drawn with food insecurity in humans. Families with limited financial resources may experience long-term, cyclical variation in food availability, with overconsumption at the beginning of the month after distribution of SNAP benefits followed by deprivation at the end of the month when benefits have been exhausted.

Another possible reason for this relationship is that low-income families may spend their limited food budget on high-calorie, low-quality products. SNAP has no regulations to influence the quality of foods purchased and pays for an estimated $4 billion in soft drinks per year, or about 20 million servings each day.6 Research suggests that if a child consumes 20 oz (600 mL) of a sugary drink, she will become hungrier more quickly than if she ate a large apple and a heaping tablespoon of peanut butter, even though both have about the same number of calories. Thus, the present lack of focus on food quality in SNAP may simultaneously exacerbate hunger and promote obesity.

The effects of food insecurity on children's physical and mental health have been well documented and include iron-deficient anemia, developmental delays, learning difficulties, impaired acquisition of social skills, and a variety of emotional and behavioral problems. Over the long term, poor diet quality among low-income children will fuel the pandemic of obesity and its related diseases.5 Without public health intervention, the economic costs arising from this pandemic may have catastrophic consequences to society as this generation ages. ... SNAP is essential for hunger prevention in the United States, but its exclusive focus on food quantity contributes to malnutrition and obesity, and is misaligned with the goal of helping beneficiaries lead healthier lives. The nation's $75 billion investment in SNAP could provide a major opportunity to reduce the burden of diet-related disease among low-income children and families if policies that promote nutritional quality are instituted.