HR is probably wrong. Here is the tax bulletin.
https://www.tax.ny.gov/pdf/memos/income/m06_5i.pdf
Generally speaking, you will file a NY non-resident return to report your NY-source income only. You will file a Kansas resident return to report all your worldwide income (your NY income plus anything else like dividends, interest and capital gains, gambling prizes, side hustle, and so on). Kansas will give you a credit for taxes you pay to NY.
NY tax rates are equal or higher than Kansas, sorry. You should have NY tax withheld, not Kansas tax. Then you would need to make quarterly estimated payments to the Kansas tax authority equal to 5% of your non-NY income, which is presumably fairly minor. The alternative is to have Kansas tax withheld, and make quarterly payments to NY equal to 6.5% of your NY income. But this method would mean you are paying about 11% in state tax up front, half of which would eventually come back to you as a large Kansas refund. (You will pay the same net tax either way, having double withholding means less take home pay up front and a bigger refund at the back end.)
NYC has a separate income tax on top of NY state income tax, but even if the company is based in NYC, you only pay NYC income tax if you physically live or work in NYC. However, this is something you will owe if, for example, you work for 2 weeks a year in NYC for training; or you fly out for in-person meetings 2 days a month; or something like that.
