Hi, welcome to Focus Fanatics?
Is the kick something you can hear? Can you physically feel movement, or does it accelerate stronger (or slower) when it kicks? If it's physical, I wonder if you might have a bad motor mount. If the engine feels like it's sputtering out, perhaps a bad fuel filter or fuel pump not delivering the fuel properly.
In regards to the oil burning, are you noticing a lower oil level on the dip stick? Or do you see where oil is leaking? I'm not familiar with the 1.6, but the 2.0 Zetec have a common issue of valve cover gaskets going bad.