The root meaning of narrative ‘a story’ or the telling of a story. The most recent generation of academics have adopted the word to designate the expression of an understanding or account of how the world works—an ideology—through stereotyped stories. In that sense, narrative has largely replaced the term myth used in my youth fifty years ago.
(Note that myth, from Greek, and account, from French have the same earlier sense of storytelling as narrative, from Latin!)
The root meaning of espouse is ‘marry, take as a spouse’; but it has been used figuratively since the 17th century to mean ‘attach oneself to, adhere to, adopt as one’s own’ It is not at all uncommon to read of an artist or politician espousing a particular aesthetic or political theory.
So it does make sense to speak of a narrative (a way of describing events) being espoused (adopted or embraced) by a political movement or institution, as when the work you cites speaks of “the narrative espoused by terrorist organizatons”.
But the expression “espoused narrative”, although grammatically unimpeachable, is grating. It pushes the ‘espouser’, the agent whose act of personal commitment gives the metaphor meaning and force, to the margin; and it leads me to suspect that the handful of people who use the term have no understanding of the metaphor and think espouse is simply a fancy way of saying ‘accept’.
I recommend that you avoid using this catchphrase, which as jwpat7 tells you sounds like “jargony babble”.