With this budget you could buy a fully prepared second-hand competition car.
If you want to go circuit racing you could contest a series at your local venue. Cars cost from £4000 upwards, while race entries start at £200.
For example the 750 Motor Club has 14 series and championships, ranging from from saloons to single-seaters, which can be tackled within this budget, including the cost of your car. Meanwhile the BRSCC organises the Mighty Minis championship which has an estimated first-year budget of between £9,150 and £9,500.
If you perfer Rallying, Vauxhall Novas, Peugeot 106s and the like are available second-hand for around £2,500 to £4,000. Entries for single-venue events on private land start at about £200 while forest events such as the BTRDA Gold Star series are more expensive at £450 upwards per event. But don't forget that in Rallying there is a co-driver to share the costs with.
Within its championship the BTRDA runs the Rally First Series. This is specifically for cars with limited modifications, which helps to keep costs down and entry fees are also lower at around £350 per event.