I assume that the noise lines are really clipped tonal.
one way of reconstructing the signal is to use an interpolating fft (fft length > window length) so that tonals are limited to a few frequency bins. Then replace amplitude and phase by, say a linear interpolation of the adjacent (good) frequency bins. on return to time domain you should have a good approximation to the clean signal.
Edit: If you take a time bin of a (complex) spectrogram or output of FFT, then you will see the multi-harmonic lines peaking up very sharply, replacing completely the spectral information of your signal. If your signal is visible at lower and higher frequencies, then the noise, you could simply mask your interferences. this would correspond to a notch filter. you can also replace the interferences with a value that is between the upper and lower frequency bin. You do this either for amplitude and (unwraped) phase or for real and imaginary of the complex spectrum. This will not reconstruct the true signal spectrum but is the best you can do.