In this paper we focus on the so called ''identification problem'' for a backward SDE driven by a continuous local martingale and a possibly non quasi-left-continuous random measure. Supposing that a solution $(Y,Z, U)$ of a backward SDE is such that $Y_t = v(t,X_t)$ where $X$ is an underlying process and $v$ is a deterministic function, solving the identification problem consists in determining $Z$ and $U$ in term of $v$. We study the over-mentioned identification problem under various sets of assumptions and we provide a family of examples including the case when $X$ is a non-semimartingale jump process solution of an SDE with singular coefficients.