  Given an irreducible projective variety $X$, the covering gonality of $X$ is
the least gonality of an irreducible curve $E\subset X$ passing through a
general point of $X$. In this paper we study the covering gonality of the
$k$-fold symmetric product $C^{(k)}$ of a smooth complex projective curve $C$
of genus $g\geq k+1$. It follows from a previous work of the first author that
the covering gonality of the second symmetric product of $C$ equals the
gonality of $C$. Using a similar approach, we prove the same for the $3$-fold
and the $4$-fold symmetric product of $C$. A crucial point in the proof is the
study of Cayley-Bacharach condition on Grassmannians. In particular, we
describe the geometry of linear subspaces of $\mathbb{P}^n$ satisfying this
condition and we prove a result bounding the dimension of their linear span.
