We present the first direct search for lepton flavour violating muon decay mediated by a new light particle X, $mu^+ to mathrm{e}^+mathrm{X}, mathrm{X} to gammagamma$. This search uses a dataset resulting from $7.5times 10^{14}$ stopped muons collected by the MEG experiment at the Paul Scherrer Institut in the period 2009--2013. No significant excess is found in the mass region 20--45 MeV/c$^2$ for lifetimes below 40 ps, and we set the most stringent branching ratio upper limits in the mass region of 20--40 MeV/c$^2$, down to $mathcal{O}(10^{-11})$ at 90% confidence level.