The best way to get the appropriate page to rank for any particular query is to target that page to what users are actually looking for.
Let's take your example of "New York". You have two pages for it:
New York
New York is the third largest state in the US. It has...
New York
New York is the largest city in the US. It has...
You say you want your state page to rank better when the user searches for "New York". You think that this would be appropriate because New York State is larger than New York City. However, Google ranks pages for a query based on user intent. Some users that search for "New York" may be looking for information about the state, but many more are probably looking for information about the city. Still other may be looking to find out why the two are named the same thing. Google usually ranks the page that more people are looking for first. It uses click through rate "CTR" to determine which of the pages on your site actually match the query better. If people are looking for the NYC page, Google tries to make them happy.
You can help by differentiating your pages. Especially the page titles:
New York State (NY) Geography - MyBrand
New York is the third largest state in the US. It has...
New York City (NYC) - MyBrand
New York, NY is the largest city in the US. It has...
Clarifying in the title what the page is about really helps users. They won't have to click into the page to find out if it is the one they are looking for. They will know before they click.
Putting full information into the titles can also help the pages rank for a variety of queries. There are a ton of things that users may actually be searching for that your page could answer:
- new york
- new york city
- new york state
- ny
- nyc
- new york, ny
- new york geography
- new york size
- new york mybrand
Google is generally very good about getting the correct page to come up for each of these queries when you describe the page in the page title appropriately for users. When Google is getting it wrong, think about what users are actually looking for and try to describe it better for them so they click on the right thing. In many cases user intent may not be clear. In some cases it might be an opportunity to write a new article addressing any confusion:
Why Do New York State and City Have the Same Name?
New York was named in....