No, such engines do not exist, as is implied in the question already:
I can't tell if this idea is good or simply bad, nor I could find any examples of such engine.
Pic above is an own photo, used earlier in this question-and answer about the efficiency of turbofan engines. In such an engine, turbines power compressors mounted on the same shaft - kinetic energy is transmitted directly, with only the fluid friction losses in the bearings yielding some efficiency reduction, as explained in @mins answer.
Powering the compressor with electrical power would not solve any problem, and introduces many others:
- That is a lot of power that needs to be created by an additional electrical pump. Even without the batteries (what would they be for?) adding a lot of weight.
- Transmission losses. Converting one form of energy into another always comes at some loss in the form of heat. With the proposed set-up, mechanical energy is transmitted into electrical energy into mechanical energy into compression energy. With induction losses in the electric cables, heat loss in the electromotor of the compressor, friction losses in the pneumatic tubing. Losses, losses, losses, for what gain?
- Extra failure modes, as @MichaelHall mentions. There may not be any catastrophic loss from system failure. Fewer systems means fewer failure rates.
The only problem that sometimes needs to be fixed with turbine engines is in rotational energy transmission - gearing. The rpm's of the axes are very high, which particularly for large turbofans creates a blade tip speed imbalance. The long fan blades impose a tip speed limit that is slightly supersonic, the much shorter turbine blades must make do with much lower tip speeds. An actual problem looking for a solution:
- In aircraft turbofans (described in this answer), multiple concentric axes can be mounted, and these axes are not mechanically coupled to each other. Allowing a lower rpm for the longer fan blades.
- In aircraft turboprops and turboshafts, the thrust generating blades are even longer, requiring mechanical gearing. Introducing weight & friction losses, for a gain in propulsive efficiency.Pic below: mechanical gearing contained in the light green frame at the front..
- In trains and ships where the earth carries the weight without induced power loss, turbo-electric transmission can take the place of the mechanical gearbox. Particularly useful if the fast spinning turbine is running while the output propulsive means in stationary. The train wheels in the station, the battleship propellers in the harbour. Note that in this set-up the turbine engine is fully mechanically coupled on the drive shafts, it is only the output of the engine that is converted into propulsive energy.
When does the battery stack pose a solution and not only a problem? In hybrid vehicles, converting deceleration into electrical power (brake heat is fully lost kinetic energy). In ships and trains, enabling silent departure and starting the noisy turbines further away. Otherwise: no problem to fix, which is why any examples of the engines described in OP could not be found.