  A study of $B^+ \rightarrow J/\psi \eta K^+$ decays, followed by $J/\psi
\rightarrow \mu^+ \mu^-$ and $\eta \rightarrow \gamma \gamma$, 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$^{-1}$. The $J/\psi\eta$ mass spectrum is investigated for
contributions from charmonia and charmonium-like states. Evidence is found for
the $B^+\rightarrow \left( \psi_2(3823) \rightarrow J/\psi \eta \right) K^+$
and $B^+\rightarrow \left( \psi(4040) \rightarrow J/\psi \eta \right) K^+$
decays with significance of 3.4 and 4.7~standard deviations, respectively. This
constitutes the~first~evidence for the $\psi_2(3823) \rightarrow J/\psi \eta$
decay.
