  We study the spreading of single-site excitations in one-dimensional
disordered Klein-Gordon chains with tunable nonlinearity $|u_{l}|^{\sigma}
u_{l}$ for different values of $\sigma$. We perform extensive numerical
simulations where wave packets are evolved a) without and, b) with dephasing in
normal mode space. Subdiffusive spreading is observed with the second moment of
wave packets growing as $t^{\alpha}$. The dependence of the numerically
computed exponent $\alpha$ on $\sigma$ is in very good agreement with our
theoretical predictions both for the evolution of the wave packet with and
without dephasing (for $\sigma \geq 2$ in the latter case). We discuss evidence
of the existence of a regime of strong chaos, and observe destruction of
Anderson localization in the packet tails for small values of $\sigma$.
