Many years ago it was possible to get a discount by paying by cash, rather than using 'hire purchase'. (My father did it with most big purchases). That was because each shop had its own hire purchase scheme, meaning that they'd only get the money in dribs and drabs if you paid that way. It was better for their cash flow if a customer paid cash.
Nowadays the opposite is true. If you pay by credit in a store, the store gets the full amount straight away plus a commission from the credit firm which actually provides the credit. (So most retailers actually prefer customers to pay on credit).
However, if you're not even referring to the possibility of any credit arrangement then, to all intents and purposes, a debit card counts as 'cash' anyway. Unlike using a credit card (where the retailer is charged a percentage of the sale price), debit card transactions attract a fixed fee (of around 20p) irrespective of whether the sale is for £1 or £1m.
So offering to 'pay cash' isn't likely to be attractive to a retailer.
The only time when paying cash is likely to get you a discount is when using the services of a VAT-registered tradesman (such as a plumber or back street garage). Some such tradesmen will offer to let you pay the non-VAT price for the job in cash, so that they can then fail to enter the transaction in their books (saving them from paying tax on their profit). It's not legal but it's a good way of saving money ;-)
Chris