I have not worked in the Food Stamp, Medicaid, or TANF programs directly since 2005, but I checked the online policy handbook for Texas Health and Human Services Commission, and the policies are basically the same in this regard. Most illegal aliens tend to have intact families, and so usually they would not be on TANF (cash assistance), but their children, if born in the United States, are usually eligible for Medicaid, and their children are eligible for Food Stamps. Not only that, the fact that the parents are "not eligible" for Food Stamps, often results in them getting more Food Stamps for their household than they would have if they were eligible -- because instead of counting their income fully, their income is prorated.
I remember when the Amnesty Aliens under the Reagan Amnesty completed their five years of "ineligibility" in the early 1990's. Back in those days my Spanish was better than it is now, but still limited, however, I could usually get through most of an eligibility interview before calling in a translator to make sure that something wasn't confused. And if I was dealing with someone who could speak some English, usually I could get through the whole interview. On one occasion I was interview a man who was a blue collar worker, making decent money, who was married, and had about 5 kids. He had been getting several hundreds of dollars each month in Food Stamps, despite him and his wife being "ineligible". I made it through almost the entire interview, but when I ran the budget on the case, now that he was "eligible" and his income was counted fully, his case was denied. So I tried to explain to him that now that he was "eligible" he was ineligible for anything. He was understandably confused. Finally he said, "Somebody speaka da Spanish?" So I brought in an interpreter, but it didn't make any more sense in proper Spanish. Welcome to the United States of America!
It makes no sense that a household should end up with more benefits because the wage earner is "ineligible" than they would get if he was "eligible". What should be the policy is that the benefits should be calculated for the household the same way as they are for a fully eligible household, but then the benefits should be divided by the number of people in the household, and multiplied by the number of "eligible" people. And so, for example, if the whole household would otherwise be eligible for $400.00 per month, and there are two ineligible adults and 5 eligible children, it should be calculated thus: $400/7*5=$286. But in the case that I mentioned above, since his whole household would have been ineligible, it should have been calculated as $0/7*5=$0 all along.