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

Yes. Because of daylight saving time the gap moves between +12h and +13h.

When is the best time to call New York from Hong Kong?

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

Hong Kong is 12h ahead of New York. This gap is not fixed: daylight saving time moves it between +12h and +13h during the year.

Hong Kong

Hong Kong SAR China

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New York

United States

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

City	Time zone	UTC offset
Hong Kong · Hong Kong SAR China	Asia/Hong_Kong	UTC+8
New York · United States	America/New_York	UTC−4

What is the time difference between Hong Kong and New York?: Hong Kong is 12h ahead of New York. This gap is not fixed: daylight saving time moves it between +12h and +13h during the year.
Does the time difference between Hong Kong and New York change during the year?: Yes. Because of daylight saving time the gap moves between +12h and +13h.
When is the best time to call New York from Hong Kong?: There is no overlap between standard working days, so one side has to call early or late.