To be honest I don’t know if there is the problem actually. Most important thing is that your notebook starts. I don’t know if this is so important if this happen after 5 or 10 seconds.

We don’t know which notebook model do you have, how old it is, since when you have noticed this strange issue, so it is not easy to understand the problem.

It is not easy to understand this statement. Does it mean that when you click on restart or shutdown option in start menu notebook react properly but cannot restart or be switched on immediately?

From my knowledge if the internal temperature is too high, the notebook refuses to boot up straight away after shut down… maybe this is the key…
Try to clean the cooling fans using compressed air spray.

Regarding the BIOS update:
Usually the BIOS update should run smoothly and is not dangerous if you follow the instruction. But not quite sure if the BIOS update would help…