Despite what you say about the glow plugs the symptoms do sound like there's a fault here.

When the engine is cold, the fuel does not vapourise properly, so ignition is difficult. When it does fire, there will be a lot of excess fuel in the chambers, hence the smoke. Once the engine is warm, no more vapourisation problems, and it starts fine.

Have you checked for power at the glow plugs when you initiate the glow cycle?

The difference in clicks will be down to the relay.

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Cheers, Neil