Does the time difference between Karachi and New York change during the year?

Yes. Because of daylight saving time the gap moves between +9h and +10h.

When is the best time to call New York from Karachi?

There is no overlap between standard working days, so one side has to call early or late.

Karachi is 9h ahead of New York. This gap is not fixed: daylight saving time moves it between +9h and +10h during the year.

Karachi

Pakistan

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New York

United States

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A 9-to-5 working day in Karachi does not overlap one in New York at all — somebody has to take the call outside office hours.

City	Time zone	UTC offset
Karachi · Pakistan	Asia/Karachi	UTC+5
New York · United States	America/New_York	UTC−4

What is the time difference between Karachi and New York?: Karachi is 9h ahead of New York. This gap is not fixed: daylight saving time moves it between +9h and +10h during the year.
Does the time difference between Karachi and New York change during the year?: Yes. Because of daylight saving time the gap moves between +9h and +10h.
When is the best time to call New York from Karachi?: There is no overlap between standard working days, so one side has to call early or late.