I know most scales/modes like Ionian (major), Dorian, Lydian, Mixolydian and Aeolian (natural minor) have a major second note called the supertonic, but some scales/modes like Phrygian and Locrian have minor seconds above tonic making me wonder if there is a specific name for a lowered supertonic note.
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2 Answers
It's still called the supertonic. It might be called the "flatted supertonic" if being discussed in relation to another mode.
"Supertonic" refers to the second note of a diatonic scale, independent of the interval between it and the tonic.
In the spirit of the accepted answer of Other than scale-degree 7, do other scale degrees have names for their altered forms?, one may choose to refer to this as the hypo-supertonic, but I've never heard that term used, and would frankly advise against it.
Colloquially, plenty of musicians refer to this as "flat 2" or "the Phrygian 2" (the latter of which also applies to the Neapolitan chord, ♭II).
I've also heard "upper leading tone" used to refer to anything a half step above a pitch of resolution. Thus in minor ♭6 could be an "upper leading tone" to scale-degree 5, and I've heard this lowered 2 described as an upper leading tone to 1.