In my experience (military), there are two times engines are "cleaned";
The first being a de-salination of the compressor and turbine blades, where they are washed with fresh water for a minute and the engine is then vented for a further 5 minutes to expel any excess water which could hamper ignition on start-up and also cause flame-out.
The second instance, is a wash with an aqeous detergent which is again, pumped into the engine via an external rig for one minute, it is then left to soak for 20 minutes and then washed clean for a minute as with the previous instance, and the engine is then vented prior to start-up. Flame-out occurs more frequently with this option.
In terms of fuel economy...I doubt it would save that much to make any real discernable difference, the de-salination and industrial washes are done for the cleanliness of the engines, and they are also boroscoped every couple of flying hours to check for any damage to the blades etc.
Hope this helps.