This paper presents weighted approaches for integrated active noise control and noise reduction in hearing aids. The unweighted integrated active noise control and noise reduction scheme introduced in previous work does not allow to trade-off between the active noise control and the noise reduction. In some circumstances it will however be useful to emphasize one of the functional blocks. Changing the original optimisation problem to a constrained optimisation problem leads to a scheme based on a weighted mean squared error criterion that allows to focus either on the active noise control or on the noise reduction. It is similarly possible to derive a scheme that allows to focus either on reducing the speech distortion or on reducing the residual noise at the eardrum. In a single speech source scenario and when the number of sound sources (speech plus noise sources) is less than or equal to the number of microphones, it is possible to derive a simple formula for the output signal-to-noise ratio of the latter scheme. It can then be shown that this scheme delivers a constant signal-to-noise ratio at the eardrum for any weighting factor.