From a notational perspective, is it clear that the second C4 in the image below is flat (if in fact it is)? I've never encountered a situation like this before.
Yes, it's C♭.
The standard definition of an accidental is that, assuming only one voice in a staff, the accidental modifies the specified pitch in that register and in that register only.
In other words, the C4 at the end of beat 3 is already lowered on account of the C♭4 on beat 2 of the left hand. The clef change makes absolutely no difference, because the C in that register has already been lowered to C♭4.
First, harmonically, it is most reasonably flat, so if that is the only thing stumping you, be pretty well assured that it is flat. Second, notationally, as it stands, with no other harmonic context, I think you could find it argued either way. There is, in the left hand, a preceding C-flat, so regardless of clef, your hand has already played the note there, and the author or arranger may assume the performer will understand that the flat persists. Otherwise, to be unequivocally clear, he could've added a courtesy accidental. I believe you are well in the clear to read that as C-flat.