First, the answer depends on whether your HSA contribution is an "employer contribution" (shown on your W-2 in box 12 with a code of W) or a "personal" (direct) contribution.
The "employer contribution" (which includes both what your employer contributed and what you contributed through payroll deduction) is part and parcel of your W-2. This is because the code W amount is removed from Wages in boxes 1, 3, and 5 before the W-2 is printed. Thus, when you use the Wages number, you are automatically including this type of HSA contribution.
So in the case of having one job in one state and another in another state (I don't know if this is what you did), when you enter the Wages from the two W-2s, you have already allocated this type of HSA contribution to the correct state.
Likewise, if your HSA contributions are scattered more or less evenly over the course of the year (as it typical with the "employer contribution", then allocating this type of contribution by months lived in each state is a good approximation.
The other type of HSA contribution is the "personal" or direct contribution. This is the contribution that it found on the Massachusetts and New York tax returns.
This type of contribution can be a one-time contribution, made on a specific date and allocated to the income belonging to a particular state. I assume that this is the type of contribution that you made.
As you note, the New York program allows you to allocate only the "personal" HSA contributions that belong to particular state's source income to that state. However, the Massachusetts program does not appear to allow this; instead the Massachusetts program takes the "personal" HSA contribution and prorates it by the number of months the taxpayer lived in that state, and provides no way to change this.
Leaving this aside, the way to deal with this is to make sure that you don't over-deduct the "personal" contribution.
When you go through the Massachusetts interview in TurboTax, you will eventually see a screen that shows you (among other things so look carefully) the amount of the Massachusetts HSA deduction (it should be the prorated amount of the amount on line 12 on Schedule 1 of your 1040). Take the prorated amount and subtract it from the full HSA "personal" contribution to get the New York State amount.
Then go to the New York State interview and enter that amount on the screen that asks for your New York portion of the HSA deduction.
Yes, this is allocating the "personal" contribution to both states, but so long as you have also correctly allocated the Massachusetts source income and the New York source income to the two states, the results will be work out. You will deduct the "personal" contribution in one place or the other, so as long as the state tax rates are similar, the effect on you will be minimal.
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