Rather than use grit, I'd add some sharp sand, it all helps to break up the clay and aids drainage, it's also cheaper and you can get a massive bag from a builder's merchant. Carrots in particular love free draining soil, so add plenty of sand (a 50/50 mix) to the area you want to grow carrots in. Any compost you can add to the soil (even if it's cheap grow-bags) will improve the soil, and like others have suggested, well rotted manure, most stables will be very happy for you to take it away! The best time to do it is autumn/winter, as the cold helps break down any big lumps, and it you keep raking it, you will have lovely soil next year.
In the meantime, why not try putting things in planters? Most stuff will happily grow in large pots, some things are better (in my opinion, strawberries do better in pots as they're easier to protect from slugs - they are every fruit and veg growers nemesis). Have a look around places like Aldi, poundstretcher etc, they often have very big pots at reasonable prices.