First, if you work remotely from NY, you shouldn't need to pay Hawaii tax, ask your employer why you need to pay that.
Second, in order to not be double-taxed, most states give credit for taxes paid to another state on the "shared" income both states claim. This is not what you should need to enter, the TurboTax program should carry the tax over to the New York return. The credit is always at that state's rate. If Hawaii has a higher tax rate on the income, New York will only give credit for the amount they would have charged as tax.
Usually this is accomplished by doing the non-resident or part-year resident return FIRST in TurboTax, then the resident return. The tax you pay on the first return (Hawaii) is used to figure the tax (and the credit) on the second return (New York).
When you do the non-resident Hawaii return, what is the result? Does TurboTax say you get all the Hawaii tax refunded?
If yes, why does Hawaii think you should pay Hawaii tax?
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