  For an appropriate choice of a $\mathbb{Z}$-grading structure, we prove that
the wrapped Fukaya category of the symmetric square of a $(k+3)$-punctured
sphere, i.e. the Weinstein manifold given as the complement of $(k+3)$ generic
lines in $\mathbb{C}P^2$ is quasi-equivalent to the derived category of
coherent sheaves on a singular surface $\mathcal{Z}_{2,k}$ constructed as the
boundary of a toric Landau-Ginzburg model $(\mathcal{X}_{2,k},
\mathbf{w}_{2,k})$. We do this by first constructing a quasi-equivalence
between certain categorical resolutions of both sides and then localising. We
also provide a general homological mirror symmetry conjecture concerning all
the higher symmetric powers of punctured spheres. The corresponding toric
LG-models $(\mathcal{X}_{n,k},\mathbf{w}_{n,k})$ are constructed from the
combinatorics of curves on the punctured surface and are related to small toric
resolutions of the singularity $x_1\ldots x_{n+1}= v_1\ldots v_k$.
