  We study the SYK$_4$ model with a weak SYK$_2$ term of magnitude $\Gamma$
beyond the simplest perturbative limit considered previously. For intermediate
values of the perturbation strength, $J/N \ll \Gamma \ll J/\sqrt{N}$,
fluctuations of the Schwarzian mode are suppressed, and the SYK$_4$ mean-field
solution remains valid beyond the timescale $t_0 \sim N/J$ up to $t_* \sim
J/\Gamma^2$. Out-of-time-order correlation function displays at short time
intervals exponential growth with maximal Lyapunov exponent $2\pi T$, but its
prefactor scales as $T$ at low temperatures $T \leq \Gamma$.
