= Solution
With the PDF's inverse <Fourier transform> convention, the derivatives act as
$$
\partial_t\mapsto-i\omega,\qquad\partial_x\mapsto-ik,
$$
so its forced differential operator becomes $\omega^2-k^2+i\gamma\omega k^2$. Thus
$$
\boxed{\widehat G(k,\omega)=\frac1{\omega^2-k^2+i\gamma k^2\omega}.}
$$
For the physical impulse response choose the <retarded fundamental solution>, vanishing before $t=0$. For $k\ne0$ and $\gamma>0$, its poles lie in the lower half-plane. The <retarded Fourier pole prescription> $\omega\mapsto\omega+i0$ specifies the zero mode and the zero-damping limit as well; a Green function without a temporal or boundary prescription would otherwise permit homogeneous additions.
This is the <Kelvin-Voigt rod Green function>. Multiplying the homogeneous real-space equation by $-1$ gives
$$
\phi_{tt}-\phi_{xx}-\gamma\phi_{txx}=0.
$$
A spatial <Fourier mode> therefore satisfies $\ddot\phi_k+\gamma k^2\dot\phi_k+k^2\phi_k=0$: positive $\gamma$ really damps the motion. For sufficiently decaying fields its <energy> satisfies
$$
\frac d{dt}\frac12\int(\phi_t^2+\phi_x^2)\,dx=-\gamma\int\phi_{tx}^2\,dx\leq0.
$$
This checks the sign of the imaginary term in $\widehat G$ independently of the transform calculation.
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