Where did British troops hope to separate New England from the Middle States?
The British planned to cut New England off from the rest of the colonies along the Hudson River.
The British controlled New York city and Harbor on the Hudson River. The plan was for a force to drive northward along the Hudson out of New York.
Two other armies were to drive south from Canada which was also controlled by the British. One army mainly of Allied Native American Indian forces was to drive east and west along the Mohawk linking up with the other army mainly of British regulars driving south.
The link up of the armies never happened. The army in New York never started North. The Indian army from the west got stalled at forts along the Mohawk. The last army was defeated at the battle of Saratoga and surrendered.
The surrendered at the battle of Saratoga was instrumental in the French thinking that the colonies had a chance of winning against the British. This resulted in the French involvement in the war and the final victory with French help at Yorktown.