As someone who volunteers at our local foodbank and has a pretty good grasp on the challenges some families face regarding food - let me say that your statement above is correct.
Let me follow-up and say however that I do no believe that a tax on high fat foods, sugar, snack foods, etc. will help the problem - this will only cause more stress on low income families facing food challenges. It is unlikely that any tax will be high enough to make the healthy food look cheap in comparison. The tax will not change the food habits of families, it will simply put more stress on low income families who will be unable to feed themselves.
A better path (that has been experimented with) is to hand-out discount coupons for healthy food for families on SNAP. The coupons allow families to buy particular healthy food items at deep discounts (SNAP card required). This type of program has a better impact on improving the nutrition of low-income families while not increasing their food stress.
That's a good idea, coupled with higher health taxes on unhealthy food might also encourage healthier choices. If not then they get to help with their own health care cost. Win/win