 Solution

A meromorphic function on a compact Riemann surface has finitely many poles, say $p_j$ of orders $r_j$. For each $p_j$, choose a projective line through it that does not contain $C$. Its hyperplane divisor contains $p_j$ with multiplicity at least one. Include $r_j$ copies of this divisor. The sum then dominates the pole divisor of $f$.

Add further arbitrary hyperplane divisors until their number $m$ satisfies $m\ge d$ and $md>2g-2$. Adding effective divisors cannot destroy the pole bound, so for $D=H_1+\cdots +H_m$ we haveFor a constant function, including zero, no initial poles need to be covered; simply add sufficiently many lines.