The process $e^+e^-{to}K^+K^-eta$ has been studied in the center-of-mass energy range from 1.59 to 2.007,GeV using the data sample of 59.5 pb$^{-1}$, collected with the CMD-3 detector at the VEPP-2000 $e^+e^-$ collider in 2011, 2012 and 2017. The $K^+K^-eta$ final state is found to be dominated by the contribution of the $phi(1020)eta$ intermediate state. The cross section of the process $e^+e^-{to}phi(1020)eta$ has been measured with a systematic uncertainty of 5.1$%$ on the base of 3009 $pm$ 67 selected events. The obtained cross section has been used to calculate the contribution to the anomalous magnetic moment of the muon: $a_{mu}^{phieta}(E<1.8, {rm GeV})=(0.321 pm 0.015_{rm stat} pm 0.016_{rm syst}) times 10^{-10}$, $a_{mu}^{phieta}(E<2.0, {rm GeV})=(0.440 pm 0.015_{rm stat} pm 0.022_{rm syst}) times 10^{-10}$. From the cross section approximation the $phi(1680)$ meson parameters have been determined with better statistical precision, than in previous studies.
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