I have had a very similar budget before, particularly when on maternity leave. Even now we tend to average £80 per week at the supermarket for two adults and two children. So it is doable but I'd be trying to stick to the £50 a week budget rather than £40.
We eat mostly vegetarian meals and I usually only buy fresh meat if it has a reduced sticker. This week I got two large trays of chicken drumsticks on yellow sticker for £2 each and they went straight into the freezer to use as we need them.
I make huge use out of our freezer and have a second one in the garage. I prioritise freezing meat and fish, bread products, fruit and vegetables.
Most of our vegetables are frozen. They are often just as nutritious as they haven't been sat in a warehouse for weeks. It also means I can have lots of variety without any food waste as it's easy to grab a handful of frozen carrots, broccoli, spinach, leeks, green beans, sweetcorn, peas etc and put them in a meal. Our fresh vegetables tend to be cheap ones that last a long while e.g. potatoes, onions, cabbage, peppers etc.
For fruit we tend to stick to the cheaper varieties. Bananas, apples, satsumas, pears, plums etc. I would only buy berries frozen as they're much cheaper that way and again no waste as fresh go off quickly. We have three rhubarb plants in our garden that I planted years ago and give us plenty of rhubarb between April and July that again I chop up and freeze. We also pick blackberries every year.... I must have at least a kilo of blackberries in our freezer at the moment as there are loads that grow round us. So that cost me nothing other than a bit of time.
I freeze most of our bread and toast it straight from frozen to avoid wasting anything. If you see a reduced loaf or anything like hot cross buns, slice them up and put it in the freezer.
Breakfasts tend to be a rotation of porridge with fruit, greek yoghurt with fruit, Weetabix (own brand) or a slice of toast with peanut butter.
Lunches are things like soup, omelettes, tomatoes on toast, mushrooms on toast or crackers with cheese, hummus etc.
Some of the very cheap dinners we have are jacket potatoes, three bean chilli, casseroles, lentil daals, egg fried rice, stews with split lentils and pearl barley, pasta dishes, stir fried noodles etc.