You are in dimension $2$, which is nice, because $H^{1}(\Omega)\hookrightarrow L^N(\Omega)$ for any $N<\infty$ so you can look for a normal, weak solution. The other good news is that behind your problem is a compact operator.

Let us assume $q\in L^{\infty}\left(\Omega\right)$, $q\geq0$, $g\in H^{1/2}\left(\partial\Omega\right),$
$h\in L^{N}\left(\Omega\right)$, and
$$
\left|f\left(x\right)\right|\leq N+\left|x\right|^{N},
$$
for some $N>1.$ Then, the operator
\begin{align*}
T_q: H^{-1}(\Omega) \to &H^1_0(\Omega) \\
\ell \to & u \text{ the solution of } \\
&-\Delta u + q u = \ell
\end{align*}
is well defined and continuous, thanks to Lax--Milgram (this is where $q\geq0$ is useful) and Sobolev
embeddings (in dimension $2$, $L^N(\Omega)\hookrightarrow H^{-1}(\Omega)$).

Let $u_{0}\in H^{1}\left(\Omega\right)$ be the weak
solution of
\begin{align*}
-\Delta u_{0}+qu_{0}+h & =0\text{ in }\Omega,\\
u_{0} & =g\text{ on }\partial\Omega.
\end{align*}

Writing $\kappa:x\to kx$ your problem writes

\begin{align*}
-\Delta\left(u-u_{0}\right) & +q\left(u-u_{0}\right)=f\left(u - \kappa\right)\text{ in }\Omega\\
u-u_{0} & =0\text{ on }\partial\Omega.
\end{align*}

In other words,

\begin{align*}
u -u_0 &= T_q f\left(u-u_0+u_0-\kappa\right)\\
u -u_0 &= K(u-u_0) \\
&\text{with } K:=a\to T_q f(a +u_0- \kappa)
\end{align*}
Now $K: H^1_0\to H^1_0$ is a compact operator, as a composition of compact and continuous operators:

- The continuous map in $H^1(\Omega)$ $a\to a+ u_0-\kappa$,
- The compact embedding
$
H^1_0(\Omega)\hookrightarrow L^{2N}(\Omega),
$
- The continuous map $f:L^{2N}(\Omega)\to L^2(\Omega)$
- The compact embedding $L^{2}(\Omega)\hookrightarrow H^{-1}(\Omega)$
- The continuous map $T_q: H^{-1}(\Omega) \to H^1_0(\Omega)$.

So $u-u_0$ is a fixed point of $K$...Schauder's Fixed Point Theorem comes to mind. Alternatively, because you have positive operators and a maximum principle you can probably get away with a constructive method of iterating sub/super solutions as decribed in Evan's PDE book.

Lecture notes detailing both points of view are here, and many references are included.