Let us say a topological space $X$ is a countable union of second countable spaces if there exists a sequence of subsets $\{X_n\}$ of $X$ with $X=\cup X_n$ such that the relative topology on $X_n$'s is second countable. Clearly $X$ will be separable.

Q. What topological properties P are strictly between separabilty and countable union of second countable spaces?

Countable union of $2^{ed}$-countable spaces $\prec$ P $\prec$ Separablity