Answer:
Sphere of influence is the correct phrase.
Step-by-step explanation:
Example:
British and Russian empire designed influence in Persia.
They were not actually interested in the country itself, but were extremely interested in the position it was located on. Britain wanted to keep safe British India, so the Russian's can't expand. But the Russians have already expanded in Kazakhstan, Uzbekistan and Turkmenistan. The British weren't very happy since Persia had borders on that region.
In order to maintain peace between the two of them, they made an agreement that said that Britain would have influence on Eastern Persia, and at the same time, Russia will have impact on the Northern Persia.