Minor and major modes, which replaced in the 17th century all the older modes, do not share the same stability. Minor mode is much influenced by major mode on its ^6 and ^7 degrees. Major mode has imposed the leading note as 7th degree, so even in minor mode the 7th is most often the leading note, i.e. a semitone below tonic, not a full tone.
Now this unstability implies a similar unstability of the 6th degree, because when playing a melody with consecutive notes, it is "forbidden" (i.e. outside of western tonal style) to have an interval of 3 semitones, such as between Ab and B. So when descending from C to G, you are left with either doing C - B - A - G, or C - Bb - Ab - G. The later is more frequent than the former, and when ascending, it is the contrary: G - A - B - C is more frequent than G - Ab - Bb - C. This is the reason for the names "ascending melodic minor" and "descending melodic minor". But one can still use C - B - A - G or G - Ab - Bb - C, it depends upon context, especially chords. (And whether the notes are passing notes or not: some composers such as Bach and Mozart sometimes write a passing A above an F minor chord which contains an Ab...).
And then, some voices may still have to jump from Ab to B or B to Ab, for some chord progression. But this is usually hidden in intermediate parts, because it hurts the western tonal ear (due to the 3 semitones interval, and due to the fact that the leading note B is supposed to rise to C if C is in the next chord). Hence the name "harmonic minor" when Ab and B are being used consecutively.
So, to sum up, the C - B - A - G in C minor is perfectly correct in western tonal music, only less frequent than C - Bb - Ab - G.