I have used shredded newspaper ( black & white only) also broken down boxes that all with deteriate. Just lay in walk ways and shredded paper around the plants to hold moisture. All good for recycling instead of putting in land fills.
Yes, you can use hay. Mulch helps also. They also help hold the water in around the plants and doesn't cost a whole lot if you get it in a big bag. They also sell this black mesh net stuff that helps (you lay it down like carpet). It isn't hugely expensive either.
My mother once laid black trash bags along the rows of her gardens to keep weeds down. (Secure with rocks or stakes.) You could of course, use weed killer, but I don't personally "approve" of using weed killer type poisons on anything (it hurts our earth and us too, IMHO).
You could also spray vinegar directly on the weeds, but you have to be careful as it might (probably will) also kill the veggie plants if it gets on them. Really, you can't completely get rid of weeds without serious poisons and maybe not even then... but you can try to keep them to a minimum.
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