Call a compact topological manifold $M$ primitive if there is no Serre fibration $M\to B$ where $B$ is a CW complex of dimension $0<d<\mathrm{dim}(M)$.
Given a Kähler group $G$ does there exist a primitive non-algebraic compact Kähler manifold with $\pi_1=G$?
What if $G$ is additionally assumed one-relator?
I am not sure how one checks primitivity for a compact Kähler manifold, this seems to be too delicate property.
Intuitively speaking, very non-algebraic Kähler manifolds have very simple fundamental groups. For instance, up to a finite covering a non-algebraic Kähler surface is isomorphic to either isotrivial elliptic fibration over a curve, a torus, or a K3-surface. In the first case, such fibration always deforms to a product $E \times C$ where $E$ is an elliptic curve and $C$ is a curve of genus $g \ge 1$, in the second case it is isomorphic to $\mathbb{Z}^2$, in the third case it is trivial.
In contrast, every fundamental group of a smooth projective variety is a fundamental group of a projective surface, but most of the variety of Kähler groups arise as fundamental groups of surfaces of general type, which are no doubt algebraic.
This generalizes to higher dimensions as the following.
The main invariant for working with non-algebraic compact Kähler manifolds is the algebraic dimension. The algebraic dimension $a(X)$ is defined as the transcendence degree of the field of meromoprhic functions $\mathbb{C}(X)$ over $\mathbb{C}$. It satisfies inequality $0 \le a(X) \le \dim X$ and $a(X)=\dim X$ iff $X$ is bimeromorphic to an (analytification of) smooth projective variety (equivalently, $X$ is an analytification of an algebraic space).
Every compact Kähler manifold $X$ admits algebraic reduction. This is a meromorphic map $r \colon X \to V$ to a normal projective variety $V$ such that $r^* \colon \mathbb{C}(V) \to \mathbb{C}(X)$ is an isomorphism. Of course $\dim V = a(V) = a(X)$.
The general fiber of the algebraic reduction is known to be special manifold in the sense of Campana (see the survey https://mast.queensu.ca/~mikeroth/proceedings/Campana-Survey-Special-Manifolds.pdf and the references inside).
Campana conjectured that the fundamental group of a special Kähler manifold is virtually abelian. Up to my knowledge, this conjecture was proved in dimensions 2 and 3 (https://arxiv.org/pdf/1107.0168.pdf).
To summarize, it seems like the only impact that the "non-algebraic part" of a Kähler manifold might have on the fundamental group is virtually abelian and all the complexity really comes from the "algebraic" part. In particular, this is expressed by the conjecture that the class of Kähler groups is not wider than the class of fundamental groups of smooth projective varieties (proved by Claudon and Eyssidieux-Campana-Claudon in the case when the group admits an injective linear representation).
