Q:

Name for scale contain the notes C, D, E, F G#, A, A#

Is this an actual scale mode, like Phrygian, Locrian, etc, or is it just a constructed scale?

C, D, E, F G#, A, A# (C)

So, root, major second, major third, perfect forth, augmented fifth, major sixth and minor seventh.

A:

Yes this scale actually exits and it's a mode of A Neapolitan Minor which contains the notes:

A  Bb  C  D  E  F  G# A

This scale starting on C in particular would be called Mixolydian Augmented due to it resembling the Mixolydian scale, but one where you would create an augmented chord off the tonic chord.

C  D  E  F  G#  A  Bb  C

You can also think of this scale as a mix of major and minor where you would just not use the 5th and both types of 6ths and strictly use the minor's 7th as such.

C  D  E  F  Ab A Bb

