signal – Mathematical difference between FFT and STFT

Can a FFT be recreated by performing STFT on the smaller time intervals and then summing them up.

No.

The STFT loses no information, it’s invertible. Until that is, you take the magnitude, which you typically do for a power versus frequency versus time graph, the normal output format for such things. Then you’ve thrown away half the information, and can’t get back to either the FFT or the time domain.

You could invert a bunch of STFTs, before the magnitude operation, to get back to the time domain, and then FFT that.