Does the time difference between Auckland and Singapore change during the year?

Yes. Because of daylight saving time the gap moves between +4h and +5h.

When is the best time to call Singapore from Auckland?

Between 13:00 and 17:00 in Auckland, which is 09:00 to 13:00 in Singapore.

Auckland is 4h ahead of Singapore. This gap is not fixed: daylight saving time moves it between +4h and +5h during the year.

Auckland

New Zealand

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Singapore

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A 9-to-5 day in each city overlaps for 4h. That is 13:00–17:00 in Auckland, which is 09:00–13:00 in Singapore.

City	Time zone	UTC offset
Auckland · New Zealand	Pacific/Auckland	UTC+12
Singapore · Singapore	Asia/Singapore	UTC+8

What is the time difference between Auckland and Singapore?: Auckland is 4h ahead of Singapore. This gap is not fixed: daylight saving time moves it between +4h and +5h during the year.
Does the time difference between Auckland and Singapore change during the year?: Yes. Because of daylight saving time the gap moves between +4h and +5h.
When is the best time to call Singapore from Auckland?: Between 13:00 and 17:00 in Auckland, which is 09:00 to 13:00 in Singapore.