Nitpick: you're thinking of a ballistic missile. Cruise missiles are more like exploding fixed-wing drones. The V-1 flying bomb was a cruise missile, and the V-2 rocket was a ballistic missile.
A rocket straying off course could either smash into city under full power (thus being a cruise missile) or have its propulsion cut off remotely beforehand (thus being a ballistic missile), depending on a scenario. What kind of a correction were you making?