Is information lost due to Hanning window perturbations?
The STFT itself is invertible, which means by means of the STFT no information about the original signal is lost. 

For common implementations of the discrete STFT, we do not calculate the FFT of each possible shift of the window (i.e. every sample), but instead we have a bigger step size (of a few samples in time domain). This sampled STFT (because we evaluate the STFT(t,f) only at t=k*T, where T is the spacing between the windows) is actually a discrete Gabor Transform (of a 1D signal, most references online refer to Gabor Transform of images). In superficial terms, the Gabor transform is invertible as long as the sampling of the STFT is not too coarse. A necessary condition for invertablity (in continuous time) is that $TF\leq 1$, where $T$ is the sampling distance in time and $F$ is the sampling distance in frequency (since in continuous-time, we have infinite frequency resolution).  The invertability becomes the easier, the smaller $TF$, i.e. the finer the sampling of the STFT is done.

So, to answer your question: Despite sounding intuitive that some portion of the signal is lost, if the window is positioned at "unlucky" positions, the STFT does not lose information about the original signal, as long as it is sampled fine enough. So, the signal itself does not determine, if information is lost, but the sampling of the STFT does.