Quarter notes + 4 32nd notes + 8th note.
In a 3/8 time signature, there should be no space for the 8th note. What am I not seeing?
That seems to be a turn. It is indicated in front of the note so as to tell you to play the ornament before the pulse. Those will be played very quickly as not to take away too much from the pulse.
These are just like the cherry on the cake, embellishment, little extra ways in which music is made interesting.
The little notes in this bar (little as in "smaller noteheads and beams") are called grace notes. They don't count towards the total, and how they are played depends on the context. Your example might be played like this:
You are seeing, and counting, an ornament. See it play it, but it has no extra count in this bar.This is called a turn and is usually played so that it starts at the moment when that last quaver would be played. Here, it appears to be wanted to be played so the last quaver is on time.Sometimes these are written out in full, using smaller dots than the rest of them, sometimes just with a sign. There are various different ornaments, all being named separately, and the timing can and does vary.
Even though they are joined with beams, the 32nd notes are printed smaller than the surrounding notes. To my eye, that would suggest that they should be interpreted as grace notes. While grace notes are typically printed with a single flag and a slash, a run of such notes would make music hard to read, as would a run of non-beamed sixteenth or thirty-second notes. I would thus regard the triple beams not as an indicating that the smaller notes should each have a duration equal to one thirty-second note, but rather merely as a means of making a group of grace notes more readable.
The 32nd notes are just there to denote a flourish, almost like a pick up to the next note.