Suppose $S\subset \mathbb{P}^n$ is a smooth del Pezzo surface and $C$ is an irreducible smooth curve (you can make it rational if it simplifies the setting) such that $\mathcal{L}=\vert -K_S-C\vert $ is non-empty. To simplify you may consider $-K_S$ is very ample and let me deal with the degree $1$ and $2$ cases. Moreover suppose that $h^0(S,\mathcal{O}(\mathcal{L}))\geq 2$ (i.e. $\mathcal{L}$ is at least a pencil).

It was my understanding that by Bertini's theorem one could choose a general member $L\in\mathcal{L}$ such that $L$ is smooth (and reduced and connected). I have been told this is wrong and after going to Hartshorne (and Wikipedia and some expository paper by Kleiman that Francesco added to the comments) I am also of the opinion that it may actually be wrong, but that $L$ must be irreducible away of the base locus of $\mathcal{L}$.

However I am unable of providing a proof nor a counter-example. Does someone have an insight on this? I also suspect the base locus of $\mathcal{L}$ may actually be empty.

Edit: Originally $H$ was a hyperplane section. The question is actually motivated by 'the' hyperplane section so I have rephrased it to meet this point. Apologies for the confusion.