We study the effect of longitudinal space charge on the correlated energy spread of a relativistic high-brightness electron beam that has been density modulated for the emission of coherent, high harmonic radiation. We show that, in the case of electron bunching induced by a laser modulator followed by a dispersive chicane, longitudinal space charge forces can act to strongly reduce the induced energy modulation of the beam without a significant reduction in the harmonic bunching content. This effect may be optimized to enhance the output power and overall performance of free electron lasers that produce coherent light through high-gain harmonic generation. It also increases the harmonic number achievable in these devices, which are otherwise gain-limited by the induced energy modulation from the laser. Show Partial Abstract