One way to deal with non-zero mean noises in Kalman filtering is to introduce a bias parameter in the state vector which will be estimated as part of the whole filtering scheme. This bias, however, has to be observable. If not, there's no way to distinguish bias in measurement from actual state change.
In satellites for example, a gyroscope is used to integrate the attitude of the satellite (in the motion model). This gyroscope, however, can build some bias over time. The filter estimate is regularly corrected with some external sensors (such as star trackers which look at starts and deduce the orientation of the spacecraft). This measurement, accurate but slow, allows to correct the attitude estimate as well as the gyroscope bias.