Let $X$ be a (infinite) separable topological space and consider $C_p(X)$, the space of continuous functions on $X$ endowed with the point-wise convergence topology.

Q. I am looking for topological properties on $X$ which make $C_p(X)$ hereditary Lindelöf.

$$X=?\implies C_p(X)=\textrm{Hereditary Lindelöf}$$