  Alon and F\"uredi (European J. Combin. 1993) gave a tight bound for the
following hyperplane covering problem: find the minimum number of hyperplanes
required to cover all points of the n-dimensional hypercube {0,1}^n except the
origin. Their proof is among the early instances of the polynomial method,
which considers a natural polynomial (a product of linear factors) associated
to the hyperplane arrangement, and gives a lower bound on its degree, whilst
being oblivious to the (product) structure of the polynomial. Thus, their proof
gives a lower bound for a weaker polynomial covering problem, and it turns out
that this bound is tight for the stronger hyperplane covering problem.
  In a similar vein, solutions to some other hyperplane covering problems were
obtained, via solutions of corresponding weaker polynomial covering problems,
in some special cases in the works of the fourth author (Electron. J. Combin.
2022), and the first three authors (Discrete Math. 2023). In this work, we
build on these and solve a hyperplane covering problem for general symmetric
sets of the hypercube, where we consider hyperplane covers with higher
multiplicities. We see that even in this generality, it is enough to solve the
corresponding polynomial covering problem. Further, this seems to be the limit
of this approach as far as covering symmetry preserving subsets of the
hypercube is concerned. We gather evidence for this by considering the class of
blockwise symmetric sets of the hypercube (which is a strictly larger class
than symmetric sets), and note that the same proof technique seems to only
solve the polynomial covering problem.
