Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $C, D$ two effective divisors on $X$ intersecting at finitely many points. Is it true that if $C$ and $D$ intersect in ''low'' number of points then the morphism $$H^0(\mathcal{O}_X(-C)(d)) \oplus H^0(\mathcal{O}_X(-D)(d)) \to H^0((\mathcal{I}_C+\mathcal{I}_D)/\mathcal{I}_X(d))$$ defined by $f \oplus g \mapsto f-g$ is surjective? The question is a little vague so as to get a broader range of answer. Any reference/idea or hint will be most welcome.
P.S. If necessary assume that the sheaves $\mathcal{O}_X(-C)(d)$ and $\mathcal{O}_X(-D)(d)$ are globally generated.