Yes, you can deduct ordinary and necessary business expenses you paid with cash. You must make notes about what you did with the cash and the nature of the expenses. It's not a good practice to do on a regular basis for the reasons you cited. No receipts or proof of expense so in the future you can create a receipt and have the person sign it. Buying anything for business at a business establishment should provide you with a receipt in almost any situation when using cash. In those situations you can write on the receipt the purpose.
Enter those expenses in your business schedule when you prepare your return.
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