Second order in mass ratio radiative-recoil corrections to hyperfine splitting in muonium

Radiative-recoil corrections to hyperfine splitting in muonium of order $\alpha (Z\alpha {\left)\right(m/M)\mathrm{}}^2{E}_F$ and ${(Z}^2\alpha \left)\right(Z\alpha {\left)\right(m/M)\mathrm{}}^2{E}_F$ are calculated. These corrections are of the second order in the small electron-muon mass ratio. An analytic expression $\left[\right(-6\ln 2-\frac{3}{4}\left)\alpha \right(Z\alpha \left)-\frac{17}{12}{(Z}^2\alpha \right)\left(Z\alpha \right)]{\mathrm{}(m/M)\mathrm{}}^2{E}_F$ is obtained. Numerically the correction is equal to $-0.0351\mathrm{kHz}$ and is of the same order of magnitude as the expected accuracy of the current Los Alamos experiment to measure the hyperfine splitting.