PDR, as my colleague Autobit says, I never said it cooled 'the engine'. It cools the turbine blades, resulting in a temp drop (oh, yes I'm afraid, it DOES!)
I have personally invoked this phenomena over 1500 times in my Pegasus-propelled flying career, and watched the JPT drop almost every time (except where the water pump failed or the tank was empty) I also gave account of the increased mass flow you mentioned in your cut-and-paste reply.
So, as I stated previously, there are a combination of factors that increase the thrust in the Pegasus: the cooling effect is real (on ~700C Turbine blades, not in the chamber at >1000C!); the drop in temp permits more fuel/RPM (about 3% in fact); the higher mass flow gives higher thrust for the same RPM; and the digital engine control system re-datums to provide higher 'wet' limits for constrained times (15 seconds or much lower at times)