A turbine engine is provided that includes a turbo-compressor, a combustor section, a flow path and a recuperator. The turbo-compressor includes a compressor section and a turbine section. The combustor section includes a combustor and a plenum adjacent the combustor. The flow path extends through the compressor section, the combustor section and the turbine section. The recuperator is configured with the flowpath between the combustor section and the turbine section. An inlet duct to the recuperator is fluidly coupled with the flow path upstream of the plenum. An outlet duct from the recuperator is fluidly coupled with the plenum.