Everything already mentioned.
Melatonin
Ciobar chocolate mix, so thick you need a spoon to eat it !
Pocket Coffee chocolates ( chocolates with a shot of espresso in them)
Proper Balsamic vinegar, but beware the good stuff costs
Taralli, tiny round wheat snacks, the fennel and also chili ones are great
Porcini funghi
Extra Virgin Olive oil , look for the DOP sign meaning it's Protected Designation of Origin
Wine, look for the DOC or even better the DOCG sign, you know it's good quality then.
If buying Prosecco make sure it's Conegliano Valdobbiadene Prosecco DOCG or Asolo Prosecco DOCG
Sachets of Saffron powder, a sheet contains 4 sachets, we use it to make Risotto Milanese, I have never seen this in UK
Near the tinned fish you'll find Cantabrico Anchovies ( Spanish but everywhere in Italy) the best and so nice in a salad or with bread and butter
Chocolate, we have some wonderful brands, the Esselunga supermarket 'Top' brand is made by Domori, which is a brand that usually costs a lot. Well worth buying.
Get Pesto from the fridge section, the stuff in a jar tastes noting like proper Ligurian pesto. Brands that are popular here are Pesto di Prà or Pesto Novella, if you have an Esselunga super market where you are, their 'TOP' brand is good too. Check the label though as a lot of Italians buy it without garlic as they want to avoid garlic breath, it will say 'Senza Aglio', you don't want that one! If you are anywhere near Genoa you will be spoilt for choice with pesto, it's where it comes from. Red Sicilian Pesto Pesto Siciliano) is less common in the fridge section, well here in north Italy anyway, but still worth a look.
I'd say cheese & Salame but not sure you can take back to UK these days, although all airports sell it in Duty Free. I still take it back to UK tbh never been stopped, just make sure it is vacuum packed 😉
Marvis toothpaste, not in the supermarket but many Pharmacies sell it and lots of other goodies!