The engine starts ok but it keeps revving up and down, as though its about to conk out, although, so far, it hasn't. It has also started to blow out some white smoke? I have changed the spark plug - but that hasn't fixed it. Any ideas?

Your Briggs & Stratton has a Sprint or Classic motor on it. Every few years, the diaphragm in the carburetor must be replaced.
Briggs part #495770

1. Remove the air filter assembly. This is a good time to clean and re-oil the filter.
2. Remove the 3/8" head bolt from the front of the fuel tank.
3. Remove the 1/2" head bolt from the side of the fuel tank.
4. Gently pull the fuel tank assembly toward you and tilt it away from you. While doing this, you will need to unhook the governor wire from the throttle plate.
5. Drain the gasoline (you can use it again later)
6. Remove the 5 screws that hold the carburetor on top of the fuel tank.
7. Pull the carb straight up. Be careful not to lose the spring or strainer on the bottom of the carb.
8. Remove the diaphragm.
9. Clean the carb and tank surfaces with carb cleaner spray.
10. Open the package with the new diaphragm in it. You will find 2 pieces. One is rubber, the other is paper.
11. Place the rubber diaphragm onto the fuel tank, then the paper one. Gently reinstall the carb onto the tank.
12. Push the 5 screws back into the carb. Before you tighten any of the screws, have all 5 pushed through and wiggle the carb a little, this will allow the diaphragm to align properly. Tighten the screws.
13. Reinstall the tank/carb assembly. Make sure you hook up the governor linkage.
14. Inspect the governor spring(s). They should be connected on both ends. You should not disconnect them for this repair.
15. Reinstall the air filter, gas, prime, and GO!

Good Luck!