Can quantum fields be viewed as superpositions of classical fields?

As stated in some of the comments, the weighting functional inside the functional integral is sometimes called a wave functional. This type of functional representation can be called the functional Schroedinger representation. Please see the following review by Roman Jackiw.

For a general interacting field theory, the exact solution for this wave functional is of course unknown.But an educated guess of an approximate wave functional with possibly a finite number of free parameters may lead to useful approximate solutions.

This method was suggested by Jackiw in the above review. This variational method has been used to reproduce the perturbation results of QED and QCD. (Plesae see this article by Heinemann, Iancu, Martin, Vautherin).

The variational approach is under active research for an explanation of quark confinement please see a recent work by Vastag, Reinhardt and Campagnari. However in 3+1 dimensions it hard to work without gauge fixing which casts doubts on the non-perturbative conclusions due to the Gribov problem. In most cases the trial wave functional is taken as Gaussian.

It is very plausible that this method can be adapted to all models in 1+1 dimensions which can be solved by a Bogoliubov transformation.

Also, there is a known non-Gaussian solution to the pure Yang-Mills theory known as the Kodama state given by the exponential of the Chern-Simons functional. This solution exactly satisfies the Schroedinger functional equation. This wave functional is considered to be unphysical, however, please see the following article by Witten describing some interesting properties of this state.

A quantum field theory defined by a Lagrangian and boundary conditions may lead to different solutions each describing an inequivalent quantization. Thus, there is the possibility that these different quantizations correspond to different solutions of the vacuum wave functionals, thus the solutions to the functional Schroedinger equations are plausibly non-unique.