On many FADEC engines it's all automatic and the pilot is just instructed to monitor. That might be the case for this one. In some engines, the pilot intervening can actually screw up the automatic protections that exist and potentially make a hot start situation worse.
In Layman's terms: In a piston engine, the fuel burning (and thus the energy) is contained within a chamber. In a turbine, you're basically blowing a blast of air, trying to extract energy from that blast of air, and then using it to create power. Think about a piston engine like putting your hand on a propeller and physically turning it. Now think about trying to turn that propeller by blowing air on it. You CAN do it (that's why propellers windmill) but it will take a lot more air. You've also got the combustion aspects (turbines actually run pretty rich in the combustion chamber)
There's a lot more to it than that, of course, but that's a simplified version.
As to what you can do about it, there are things you can do to improve efficiency but there are limits. You can make lean-burn combustors, but those are harder to design and also harder to keep a stable burn. You can increase the OPR (basically the compression ratio) which tends to take a more advanced, bigger, heavier compressor. That adds to cost as well as weight. You can add turbine stages to extract more power from the exhaust. Again, more weight, more cost, etc. You're doing all this with things that are inherently difficult and expensive to produce and low demand, which means development costs are going to be amortized over a small number of sales.
In my opinion, the real key would be trying to make more compressor and turbine stages as much as would be practical. Basically try to increase your OPR and extract as much energy as possible. Lean burn combustors are really hard and I think you'd be setting yourself up for issues trying to go there.
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