Suppse $X$ is an irreducible cubic surface in $\mathbb{P}^3$ singular along a line $l_1$. Then clearly there is a plane $H$ containing $l$ such that $X \cap H = l_1^2l_2$. My question is: can $X$ contain a degree $2$ plane curve other than curves of the form $l_1l$, where $l$ is some other line ?

2$\begingroup$ Yes. $X$ is the projection of a cubic scroll $S\subset \mathbb{P}^4$ from a point $p\in \mathbb{P}^4\smallsetminus S$. This $S$ can be viewed as the image of the map $\mathbb{P}^2\rightarrow \mathbb{P}^4$ given by the linear system of conics passing through a fixed point. A general line in $\mathbb{P}^2$ maps to a conic in $\mathbb{P}^4$, which projects isomorphically to a conic in $X$. $\endgroup$ – abx yesterday