Bucking

The problem is that the engine starts up fine and runs really smooth and if you push the accelerator down nice and slowly the engines revs up great.

However if you push the accelerator down quickly it coughs and splutters (just for a moment and then revs up ok….right the way up the rev range.

This does this whether under load ie driving or just sat there on tick over.

I read in a manual, this is called bucking. 

It's a 1.8 cabby with a 2e2 carb. 

Sounds like a fuel starvation problem may be carb fuel chamber floats or fuel pump diaphragm could also be air leak on induction side or ignition timing. Let us know what carbs and pump you have got.

I've advanced the timing and adjusted the fuel mixture and its running 100% better. 

The mixture screw was all the way out so I screwed it al the way in then backed it off about three turns and now its running sweet.