Why is a very small peak with larger m/z not considered to be the molecular ion?
Without knowing more details, it is hard to guess, but at this $m/z$, it seems likely that the peak is the result of one $^{12}\ce{C}$ being substituted by one $^{13}\ce{C}$. It is more useful to assume a uniform mass of 12 for carbon when analyzing such a spectrum.

Note that with more carbons in a larger molecule, it becomes more likely that at least one of them is a $^{13}\ce{C}$. Thus, the spectra become more complicated that way.

This whole analysis goes out the window if this is not an organic molecule, though similar patterns may arise for other molecules - it depends on the isotopes and their distribution.