As a former realtor in Westchester, you need to explain a few things because your question doesn't make sense.
1. You stated "House price is $70K." Is this a co-op? A house in foreclosure? There are no houses in Westchester for $70,000. Even a foreclosed house that needs major repairs would sell for about $300,000. At $70K, it means a studio or 1 bedroom co-op in Yonkers. It's not a house.
2. You stated "20 percentage down". Why 20%? Most co-ops in Yonkers only require 10% down - not 20%. If this is a house, you can get an FHA loan which just requires 3.5%.
3. If this is a co-op, you will need to fill out an application and be reviewed by the co-op Board. Most co-op Boards do not accept a co-signer. Even if your lender approves a co-signer, the co-op Board won't.
4. You stated "Offer accepted and inspection done." Did you and the seller sign a contract yet? Why did you have the inspection done if there is no contract? An accepted offer is not a contract. You may have wasted money on an inspection fee.
5. Do you have a real estate attorney? Real estate attorneys are required to close a real estate transaction. Also are you working with a real estate agent?
6. If your 2016 income is $40K, you will not qualify to buy anything. It doesn't matter what your co-signer's income is, because lenders go by your income first. At most you will qualify for a studio or maybe a 1 bedroom co-op in Yonkers. However, as I mentioned above, the co-op Board may not approve you because of your low income.
If this is a co-op, you need to know that co-op Boards need proof that you can afford to pay the monthly mortgage + the monthly maintenance fee + utilities + other expenses. They also want proof that you have savings in case you lose your job or have a situation where you lost your regular income.