If you don't want to spend much money, buy food in supermarkets and cook for yourself. Here are some hints what to buy if you want to live the Corsean way.
If you (in my opinion) have an aptitude for sadism, you might want to try Pâté de Merle. It's blackbird pie, spiced with myrthe. Production is nowadays prohibited, but stocks may be sold out. And I bet this stuff is still being produced.
If you are away from the coasts, look for brook trout (truite).
Gigot d'Agneau: lamb haunch or lamb chop or roasted lamb. Cabri: young goat, said to taste a bit boring, better: ragout. Sanglier: wild boar, available in Autumn, after hunting season starts. Roasted or ragout, even "Terrine de Sanglier" (wild boar soup). Marcassin is young wild boar. Lapin: rabbit, served in the interior. Stufatu: meat dish made from beef and lamb with noodles and cheese, rather rare. Tripes: well, not everybody's thing, but regarded as delicacy.
On Corsica there lives a fly that puts its eggs into some sorts of cheese. At some time the cheese is full of thin threadworms that jump away if you bite in the cheese. Well, maybe you won't bite in such a cheese, but many Corse like such cheese, because the excrements of the threadworms give it a certain flavour (don't ask me, I've never tried it). You won't be able to find such things in a supermarket, of course.
Brocciu is something special: a cottage cheese, spicy and creamy. Not to be confused with Brousse, which is made from milk powder. Real brocciu is only available from October to June.
The French eat lots of white bread, available as double loaf (hard crust, nothing inside, not recommendable), as baguette (thin bar of bread, good on the way), as flûtes (thicker than baguette, our favourite, also called restaurant) and as panier (kind of flûtes with pointed ends). Wholemeal bread and toast is available in supermarkets. If you can afford it, take croissants or pains au chocolat.
Honey (miel) is a popular souvenir; there are many kinds of honey available that differ in their taste. While the dark chestnut honey has a very strong, almost bitter taste, honey from the Cargèse is spicy, and honey from the Balagne is mild.
Gâteaux Corse (Corsean cookies) is a good snack for every occasion. Every baker has his own recipe.
You'll find more things to try in your travel guide if you should decide to have a closer look at Corsica.
Our favorite is Muscat, a white wine with almost liqueur-like taste and 15% alcohol. Very mellow and characteristic, and quite cheap.
If you are in remote areas in a restaurant, you'll probably find that Pastis is the cheapest offer. It's a 45% anise spirits that's drunk diluted with water (1:5) and served with ice.
Beer is a matter apart. You'll find Kronenbourg nearly everywhere, and you'll find it to be as tasty as sauerkraut juice.
If you're hiking the mountains you'll find sources nearly everywhere. We drank from everything that did not look mouldy, and it really did not do us any bad. However, I've received a report by Torben Kloss <klost000@goofy.zdv.Uni-Mainz.de> that some of the sources in the southern part of the island now have been curbed and redirected with black hoses, whose ends are ideally suited as breeding places for bacteria. He recommends using chemical desinfectants like Micropur in these places.