A little background ...
The effect that we see with currents on waves is a Doppler shift. That is that when currents are against the waves, the waves are compressed and when the currents travel the same direction then the wave are elongated. The resulting spectrums will see the peak energy shift slightly to lower frequencies or higher frequencies.

It is important to note that currents flowing in a direction perpendicular or normal to the wave direction will have no effect on the waves. It therefore is not just the magnitude of the currents but how much of the currents are projected onto the wave direction (cosine of the angle between the respective directions).

The degree to which this Doppler shift modifies the surface waves depends on the current speed relative to the wave propagation speed. This naturally means that slow propagating waves are most effected by currents. These are the short period waves.

Effects on wave measurements:
The effects on measurements really depends on the measurement technique. The AWAC which uses AST (Acoustic Surface Tracking) is a direct measure of the surface waves and therefore its response is uneffected by the currents. However the Doppler shifted surface waves are still present and measured.

Measurements that infer the surface waves from either the orbital velocities or pressure require special attention. This is because the transfer function used for inferring the surface waves is wavenumber dependent, and it is the wavenumber that is modified by the currents. Therefore the wavenumber solution must take into account the mean current and direction relative to the wave direction.

It is PUV type of systems that infer the surface waves and are most sensitive to the effects of currents. However recall that it is the small (high frequency) waves which are most sensitive to currents and often these waves are not measurable by the PUV approach due to its frequency limitations. Therefore this has not been included in the solution.