I don't think it's the frequency that's the defining factor as to whether you save money as long as it doesn't cost to go to the shops, eg because it's in walking distance or you go on the way home from work.
After all, if you shop often, even several times a week, you could pick up lots of reduced bargains or be able to stock up on special offers. The 'extra spends' are only going to end up costing extra if you're buying luxury versions that are not in the budget, or don't use the item, because otherwise it will just keep until you do need it, and reduce spending later on.
Buying the same things once a week from the same supermarket is probably the most expensive way to shop unless you're going to Aldi or Lidl where prices are generally lower and consistent, so it's definitely worth avoiding that norm if you're able to.
Otherwise to save money, you want to be buying special offers, usually getting larger packs, eg a 4 pack of beans once a month instead of a single can once a week will work out cheaper. Also rotating round supermarkets and cherry picking offers as some things are cheaper in some supermarkets, also a lot of items are always on offer somewhere so if you keep a stock in, you never have to pay full price and save quite a lot over time.
But getting toiletries, cleaning products etc is often cheaper in Home Bargains, B&M etc so worth stocking up every so often. I don't find these items to be cheaper from Amazon usually, because they cost a disproportionate amount to transport, so is factored into the cost.
The other way to reduce cost would be to look at what you're spending the most on and swap for cheaper items. Eg swap sprouting broccoli for normal broccoli reduces the cost/kg by about 70-80%, eat less meat, frozen fish and berries instead of fresh, etc etc.