Hello Artsypants,
You should have both PA and NY taxes with held from your check.
You will file a nonresident PA tax return and pay taxes due or get a refund based on your tax liability
You will file a NY resident return and get a credit on it for taxes paid to PA and pay taxes to NY or get a refund based on your tax liability
As far as deducting the business expense on your federal return as of 2018 the IRS only allows this deduction if you are self employed, if you get paid on a W-2 then the deduction is not allowed.
You may be able to deduct the business expenses on the NY return if you meet all the qualifications. https://www.tax.ny.gov/pit/file/itemized-deductions.htm
The same applies to PA. https://www.revenue.pa.gov/PATaxTalk/Pages/Article.aspx?post=32#:~:text=Pennsylvania%20Personal%20In....
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