Suppose $\kappa$ is an infinite cardinal. Can there exist a family $\mathcal{F} \subseteq [\kappa]^{\kappa}$ such that $|\mathcal{F}| = \kappa$ and for every $X \in [\kappa]^{\kappa}$, there exists $F \in \mathcal{F}$ such that $|F \setminus X| < \kappa$?
By a diagonalization argument, I can show that such a family does not exist if $\kappa$ is regular. But what if $\kappa$ is singular? I am mostly interested in the case $\kappa = \mathfrak{c} = 2^{\aleph_0}$.
I suspect that such a family never exists, but I don't see how to prove it for singular $\kappa$. Any help is appreciated!