BOAC opines
I actually think it could make a lot of sense to keep the APU running a bit longer after departure. Who knows whether a shutdown and relight MIGHT just clear an engine that has choked and spluttered to an early grave with bird debris? We are only talking around 30-50kg of fuel I guess. It is certainly the correct procedure for flameout due to Volcanic Ash ingestion.
Very true re volcanic ash, and there's a unique reason:
The ash tends to melt and "Plate" the HPT nozzle vanes with a glass-like coating. A little buildup tends to close down the flow area of the nozzle, increasing the back pressure on the compressor(s). Compressor stall (surge) is the likely result.
Shutting down the engine for a few seconds causes shock cooling, which may fracture the unwanted glass into fine particles, possibly (partially) ridding the nozzle of the problem, and permitting an airstart.