Some of the suggestsions above seem to fit; others don't really hit the spot.
It's not just "Spoken word" because everything we say is spoken word, and doesn't imply musical accompaniment.
It's not monologue, because that (e.g. Stanley Holloway) is usually just a metric poem, and doesn't necessarily have any music going on either.
It's not just soliloquy, for the same reason.
It's not recitative, as this is sung rather than spoken.

It looks as if the term used depends on the musical genre. Talking blues, Sprechgesang, and what's the Welsh one they do with a harp? I don't know of a term to cover what Jim is asking about, etc, or one that encompasses the whole lot. But it's early days yet, perhaps someone will come up with one....

Marje