Answer
Yes. If you define f=−∂μAμ then you can write the equation in the form ∂μ∂μψ=f
This is the Klein-Gordon equation with a nonzero source (f) and can be solved via Green's function methods. Once you have the Klein-Gordon propagator* G(x) (this is derived in any e.g. quantum field theory textbook) appropriate to the boundary conditions the solution can be written as ψ(x)=∫d4x′G(x−x′)f(x′)
since Green's functions by definition satisfy ∂μ∂μG(x−x′)=δ(x−x′)
where we take all differentiations to be with respect to x.
*You need the propagator in the position space representation to write this down. It is usually more convenient to write it in momentum space; you can go back and forth using (inverse) Fourier transforms.
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