A study of $B+\to J/\psi \eta K+$ decays, followed by $J/\psi \to \mu +\mu -$ and $\eta \to \gamma \gamma$, is performed using a dataset collected with the LHCb detector in proton-proton collisions at centre-of-mass energies of 7, 8 and 13 TeV, corresponding to an integrated luminosity of 9 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. The $J/\psi \eta$ mass spectrum is investigated for contributions from charmonia and charmonium-like states. Evidence is found for the $B+\to (\psi 2(3823)\to J/\psi \eta )K+$ and $B+\to (\psi (4040)\to J/\psi \eta )K+$ decays with significance of 3.4 and 4.7~standard deviations, respectively. This constitutes the~first~evidence for the $\psi 2(3823)\to J/\psi \eta$ decay.