The value of the anomalous magnetic moment of the muon, $a_\mu = (g-2)_\mu / 2$, is a fundamental quantity in particle physics. Its most precise experimental measurement
yields a deviation of 4.2$\sigma$ from the theoretical prediction within the Standard Model. In this context, the recently proposed MUonE experiment at CERN aims at
providing a novel and independent determination of the main source of theoretical uncertainty on the muon anomaly, namely the leading order hadronic
contribution $a_\mu^{\rm HLO}$, through the study
of elastic muon-electron scattering at small momentum
transfer. The anticipated accuracy of the order of 10 ppm demands for
high-precision calculations of the relevant radiative corrections to $\mu e $ scattering, as well as for robust quantitative estimates of all possible background processes. The contribution due to the emission of a neutral pion through the process $\mu e \to \mu e \pi^0$ is here studied as a source of reducible background for the measurement of the QED running coupling constant
at MUonE, and as a background for possible New Physics searches at MUonE involving $2 \to 3$ processes, in phase space regions complementary to the ones characteristic of the elastic $\mu e$ scattering. Its numerical impact is discussed by means of the Monte Carlo event generator MESMER.