would that also then provide a cooling effect across the turbine like in a piston aircraft and enriching the mixture?
Negatory I am afraid. Turbin's do not function like Piston Engines (which can only get so much air into themselves due to the valve timing). If ya put more fuel into a Turbin then the more fire you get, the hotter it runs, and the harder it drives the Turbin and the Compressor which is attached to the other end of the shaft. Hence more gas is produced which increases all the parameters, and these readings keep on climbing until you either become Temp, Torque, or RPM limited, whichever comes first, prior to some kind of melt down or other engine destruction if you keep over fuelling it for long enough.
Take care out there. BE