How to go from wav file to spectrogram back to wav file in python?
You need to keep the phase information, which isn't included in a spectrogram.  "Spectrogram" is just the magnitude of the STFT output.

So this is conceptually the same as STFT ↔ ISTFT, which is supported in Python now:


STFT
ISTFT


The Examples section in the ISTFT link does exactly what you want, modifying a STFT and inverse transforming to reduce noise.