2004 alhambra tdi (auy engine) cold start fault?

in the morning car will fire for a spilt second then cuts out then i turn it over for about 10 to 15 seconds and it coughs in to life and runs perfectly and will start on the button and runs great.i have looked at the glow plugs then i noticed there was no 12v supply to the glow plugs when glow plug light on dash comes on so i checked the relay which had a 12v supply to the relay and a switch supply to the hold on contact but relay dose not opperate so i earthed the other side of the relay and the contact closes so i check the coolant senser which i thought may control whether the engine needed the glow plugs on to start i.e.if the engine is already hot i changed the sensor but i still have the fault there is a fuel temp sender i have been told its the coolant sender controls glow plugs so i have come to a point that i think it may be a fuel supply fault but i would like some input also i cant find any fuse on the control of the glow plug