Why did both british and american armies want to hold new york city

The British as well as the American armies wanted to hold New York City because of its strategic location and its status of an important port.

Explanation:

The Americans wanted to keep New York as its was a matter of their status. The British wanted to occupy New York so that they could bank upon its status and increase trade and communication links.

The British occupation of New York threatened the communication links between New England and the other allies which could put America at a disadvantage. Both America and Britain wanted bank and benefit from the status of New York.