Using 106~million $psi(3686)$ events collected with the BESIII detector, we measure multipole amplitudes for the decay $psi(3686)rightarrowgammachi_{c1,2}togammagamma J/psi$ beyond the dominant electric-dipole amplitudes. The normalized magnetic-quadrupole amplitude for $psi(3686)rightarrowgammachi_{c1,2}rightarrowgammagamma J/psi$ and the normalized electric-octupole amplitudes for $psi(3686)rightarrowgammachi_{c2}$,~$chi_{c2}rightarrowgamma J/psi$ are determined. The M2 amplitudes for $psi(3686)rightarrowgammachi_{c1}$ and $chi_{c1,2}rightarrowgamma J/psi$ are found to differ significantly from zero and are consistent with theoretical predictions. We also obtain the ratios of M2 contributions of $psi(3686)$ and $J/psi$ decays to $chi_{c1,2}$, $b_{2}^{1}/b_{2}^{2} = 1.35pm0.72$ and $a_{2}^{1}/a_{2}^{2} = 0.617pm0.083$, which agree well with theoretical expectations. By considering the multipole contributions of $chi_{c1,2}$, we measure the product branching fractions for the cascade decays $psi(3686)rightarrowgammachi_{c0,1,2}togammagamma J/psi$ and search for the process $eta_{c}(2S)togamma J/psi$ through $psi(3686)rightarrowgammaeta_{c}(2S)$. The product branching fraction for $psi(3686)rightarrowgammachi_{c0}togammagamma J/psi$ is 3$sigma$ larger than published measurements, while those of $psi(3686)rightarrowgammachi_{c1,2}togammagamma J/psi$ are consistent. No significant signal for the decay $psi(3686)togamma eta_c(2S)togamma gamma J/psi$ is observed, and the upper limit of the product branching fraction at the 90% confidence level is determined.
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