We need to be able to either distinguish the noise from the signal, for example by frequency band, or somehow construct the noise signal so it can be subtracted from the total.
If the noise occupies narrow bands, even within the signal band, it may be possible to notch it out while still leaving a useful signal.
If that isn't the case, can the noise waveform itself be determined? For example, in the instrumentation is it possible in a second channel to capture the noise without the signal. If this is possible, then one might subtract it from the total, with an amplitude such as to minimize the resulting signal energy.
If that is not possible, but the noise signal is a reproducible waveform, perhaps it can be determined, up to some amplitude and phase, by subtracting your available signal from the signal+noise you have, and then use the same method.
But if the noise is not reproducible, and is spread over the frequency band of the signal, I do not see how they can be distinguished.
Maybe some of the signal processing experts here could chime in?
Best regards, David