On the existence of primitive pencils for smooth curves

Let $C$ be a smooth curve with gonality $k\ge 6$ and genus $g\ge 2k^2+5k-6$. We prove that $W_d^1(C)$ has the expected dimension and that the general element of any irreducible component of $W_d^1(C)$ is primitive if either $g-k+4\le d\le g-2$ or $d=g-k+3$ and either $k$ is odd or $C$ is not a double covering of a curve of gonality $k\mathrm{/}2$ and genus $k-3$. Even in the latter case we prove the existence of a complete and primitive $g_{g-k+3}^1$.