Can we use a giant thruster on the south pole to change Earth’s orbit?

I doubt it. The earth is like a bag of jello and if you tried to thrust it = problems even at 1/1000G, and your reaction mass could not travel through the atmosphere or it would entrain a huge volume of air, and, remember, this jet would have to exceed the escape velocity by some margin as 7 miles/second would be taken from it on it's upwards path. Lets say you make a tower above the atmosphere - say 100Km high - what you gonna build it from? The toughest known materials would slump into a puddle far below 100 Km. Even a step pyramid would not work as it would cover a huge area and cost the planetary GNP 100 or more times over. It is OK as gedanken ware...

I would much rather adjust the earth's axis; let's eliminate seasonal variations. Winter sucks. Those poor misled people who live in the Northern latitudes shoulda moved to habitable climates long ago anyway. Leave that shit to the polar bears.

Unfortunately as aurizon mentioned; the earth is all squishy and there's severe difficulty applying enough force to move the planet meaningfully without unpleasant side effects.