Hi.
There isn't really any 'cheap' way of going rallying or racing. Burning petrol for fun is expensive! But you can get into motorsport at grass roots level for not alot of money.
Depends what you want to do really....if its rallying, start by contacting your local motorclub and see what they organise. They might do scatter rallies or 12 cars, and then move up to road rallying....and you can use your road going mini....with a wee bit of prep.
Racing....how bout grass track minis? From what i've seen from friends doing it it looks like real good fun. But you will need a purpose built car, a trailer and a tow vehicle of some sort...all adds up.
It really does depend on what you want to do as racing and rallying are very different beasts and a mini suited to one might not suit the other. Rally cars tend to be stronger and heavier than race orientated cars.
Also look at autotesting, autosolo's and grass autotests too. All will give you a flavour for motorsport for very little outlay. Then also sprints and hillclimbs. There are some historic road rallies up your way held during the day....look on HRCR webite at the Clubman championship.
The link below shows all sorts of road rallying in your area
http://britishroadra....com/event-map/
Also if rallying is your bag, i've started a thread on here that gives you some idea of whats involved....enjoy
http://www.theminifo...-into-rallying/
Also there are loads of very knowledgeable people on here from both the racing and rallying side of things, and hopefully one of them is in your area so can offer you some guidance.