We use Raman amplification in hydrogen-filled hollow-core kagomé photonic crystal fiber to generate high energy pulses in pure single higher-order modes. The desired higher-order mode at the Stokes frequency is precisely seeded by injecting a pulse of light from the side, using a prism to select the required modal propagation constant. An intense pump pulse in the fundamental mode transfers its energy to the Stokes seed pulse with measured gains exceeding 60 dB and output pulse energies as high as 8 µJ. A pressure gradient is used to suppress stimulated Raman scattering into the fundamental mode at the Stokes frequency. The growth of the Stokes pulse energy is experimentally and theoretically investigated for six different higher-order modes. The technique has significant advantages over the use of spatial light modulators to synthesize higher-order mode patterns, since it is very difficult to perfectly match the actual eigenmode of the fiber core, especially for higher-order modes with complex multi-lobed transverse field profiles.