You can do this!!!
You really can!

You are wiser and more creative than you know!!

If it were me, what I would do is to take my clip board or a pad and I would go around to all the cabinets and the freezer- make a full inventory of what you have. I do this at the end of every month but you can do it now. Then sit down and write out all the dinners you have all the ingdients for first- on the top of another page. At the bottom of the page list meals you have most of the ingredients for and next to them, in paranthesis, put what you would need to get to make the dish. Then list what you have on hand grouped as "snacks", "Breakfast" and "Lunch". If you have anything for beverages them list that, but when money is tight I have nothing to put there because it is water... : )
Then you can really look at it and see what you have. Often we have things on hand we have forgotten about or that we can throw together and make a wonderful meal from! I always have beans and rice on hand so think what you can make from those and make that a few times a week. You can do it!!!
$10 a week will cover milk ($2.50 a gallon, here at aldi/ $5 for 2 gallons) , if you drink it, eggs, if you use them ($1.50 at Aldi) and after buying 2 gallons of milk and the eggs I would be left with $3.50. If you can get away with just one glalon of milk for the week then you would be ahead. If you have powdered milk now is the time to make some up... so you would have $3.50. this should buy you some veggies and fruits but not much. Really look in the cabinets and see if you have any canned fruits and veggies and if you can use those. I would buy a big bag of apples anda big bag of carrots... not glamourous but would get me thru assuming I already had onions and potatoes.
Can you make your own bread?

Other thing I might consider would be too look around- do you live in a state that cashes back in soda cans? If so, cash in any cans you might have. Have any pennies or change laying around? I would roll that to get enough to get a few more fruits and veggies.

Is this your permanent budget? If so I would consider making it a bit bigger- even $10 a week more, if you can... It is ok to do the bulk of your grocery whopping in one trip but if you do that you want to make sure you are getting enough food to cover the month and if it were me I would have to hide some of the stuff from my older kids so they did not eat it lal gone by the 5th of the month! LOL!
