"when student spent 3 summer months of internship in NY, the rest in Virginia."

Is this what you have?

You don't seem to have a person who moved residency. This doesn't seem to be a part-year resident, even. You seem to have a person who worked for a NY firm, who did so with earnings while in VA (telework or remote) and then onsite in NY as a temporary assignment.

NY is taxing all of it.

"If a person is not a New York resident, however, he or she is subject to New York income tax only on income that is sourced in New York. The source of compensation income generally is where a person works; thus, a nonresident employee’s compensation from working for a New York employer in an office in New York generally is taxed by New York.

If a nonresident employee of a New York employer telecommutes from his or her home outside of New York, days spent telecommuting are generally considered days worked in New York if the nonresident’s principal office is in New York."

VA is taxing the amount earned while in VA.

For the $20k: "Every resident of Virginia, including domiciliary residents, is liable to state income taxation as a resident. This means that they are subject to Virginia income tax on their entire income, whether it came from sources in or outside of Virginia."

For the $20k that is part of the $34k: "To help prevent payment of taxes to multiple states on the same income, Virginia law provides a credit for taxes paid to another state. If any part of your Virginia taxable income is also taxed by another state, this credit may be available to you. To claim it, you will need to include the Schedule OSC and a copy of the return you filed with the other state with your Virginia income tax filing."