The Skoda Kylaq will be the perfect choice for your requirements. It has a punchy 1.0-litre turbo-petrol engine and is available in both a manual gearbox and a 6-speed torque converter automatic.
The Kylaq has compact dimensions and has a well-balanced chassis that makes the car great fun to drive. The handling is particularly impressive, and it has a good build too. The Prestige variant in manual guise will fit in your budget, while the Signature+ AT variant works if you prefer the automatic.