Many people forget that children do need something to do. If you have chosen to stay in the city, more than likely you don't want your kids running around and would like to keep their activities supervised.
Check with the local YMCA. They have many resources. If their shelter is full, they might be able to help you find another place to sleep. They also have activities that your children can participate in at a low cost. There are also many after school programs that charge you based on a sliding scale. Always check with the local churches. Churches are your best bet for any help, from food to day care, education to a shelter. In my past experience with being homeless myself, Evangelical churches have been more helpful and less questioning. But just about all the churches have ways of helping out.
Even with those activities, you will find that weekends and after the programs, your children will be bored, with several hours of daylight left. The easiest solution to this, is to hit up one of those dollar stores. They have very cheap entertainment, such as travel board games, coloring books, crayons and reading materials. Try not to overload your car however. You should have only your essentials, in the trunk which I know can be difficult in deciding. Do Not buy 6 different coloring books, buy only one for each child, allow them to share a large box of crayons, and only replace as needed. Emphasize that these items are very special, and that they have to be cared for. If one child loses or destroys a book, do not replace int immediately. Wait at least a week. The child will win and complain and get on your nerves, but you have to show that these items are not easily replaced and that it is a privilege to have such things. They will learn.
Some cities and camp grounds have the dime shower. Costs you a quarter to get in, and a dime for every minute in the shower. They might have changed since the last time I used them, and NM has them all over the dessert. Get your kids cleaned up and take them to a movie, with the coins that you have set aside. This is a wonderful treat, and something that your children will start helping out to save for. There are places that you can see a movie for under $2. Don't do this often, keep it a treat, and save up most of your money for food and motel rooms.
Remember ring around the Rosie's? Duck duck goose? Clapping games? Sure you do. Even if your children are older and find these things dumb, do them. They actually come around and enjoy these types of games. It helps keep the family active and together. And many of these you can do in a parking lot. Visiting a park at every opportunity will also keep your kids active, social and out of your hair. For a list of traditional kids games, go to this wiki page.
With the entertainment you feed your kids, try to keep them physically active, or keep their brains stimulated. Just because you live in a car, doesn't mean your children will not turn out to be a surgeon, or successful in motorcycle building. And perhaps if your lucky, both.