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The engine management light comes on to inform you that the engine is not working correctly and it stores a fault code in the ECU with information on what is wrong. You can only access this information via a diagnostic device.
If the DME (engine) ECU keeps on seeing a fault on the engine operation it then goes into limp mode to protect the engine from damage (low power or stops) when you cycle the ignition it can reset the ECU and the process starts again (depending on what the fault is).
I advise you to take your car to a Renault main dealership or specialist to have a real time diagnostic check done so a full diagnose can be given (this will have a set charge of approx 20 mins labour)
I suspect that you have a failed ignition coil pack (common failure on your model) this is causing the misfire (lack of power and judder) this will be confirmed via a diagnostic check. Coil packs are approx £20 each and your engine has 4 if one has failed it is advisable to replace all at the same time and keep the others as spares.