I have not read the entire thread, mostly because I hate it when people get flamed for things that are no one's business. I don't care if she has a trip planned. Just because some people have a harder time making ends meet doesn't mean they don't deserve a wonderful vacation. I think those people (sometimes) deserve it more than anyone.
As for her original question, I think it's doable, but it will be difficult. You have a lot of frozen meat, which will definitely help you out. However with four kids, it's going to be hard to stretch it out and keep them full. It's good that the older ones will be spending time with grandma, that will definitely help!
I would go through and try to find anyway to make a little fast extra cash. Have a yard sale! Maybe your kids have a lot of toys and/or clothes they don't use or wear anymore? Yard sales with kid stuff are big hits. If you don't have time for a yard sale, then try a consignment store. You might have to wait a few weeks before you see any money and you have to split it with the store, but it's an idea.
Here's another suggestion, but it won't be a whole lot of help for June. If you have a house with a yard, try planting a small veggie garden. You can get plants through mail order pretty cheap, and it would be a great way to add some healthy options to your menus. Try growing blueberries, strawberries, cucumbers, carrots, tomatoes... And it might be a fun project for your kids to work on to take some of the responsibility off of you. But like I said, doing that won't help you out at all for this month and it's going to take a few months (depending on the size of the plants you start out with) to actually get a good bit of use out of it. But doing this could help you down the road with future months that could be tight.
I'm sure others have suggested you start couponing. I love it when I can stack a store coupon with a manufacturer's coupon and apply it to something that's on sale! It can be like a little game to see how many items I can get for nearly nothing. My favorite website is SouthernSavers.com. She will go through the local sales ads and match the coupons up for you with the sales. She will even provide you with links to printable coupons, if there are any, for sale items.
Good luck to you!!