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

Yes. Because of daylight saving time the gap moves between +11h and +12h.

When is the best time to call Lagos from Auckland?

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

Auckland is 11h ahead of Lagos. This gap is not fixed: daylight saving time moves it between +11h and +12h during the year.

Auckland

New Zealand

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Lagos

Nigeria

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

City	Time zone	UTC offset
Auckland · New Zealand	Pacific/Auckland	UTC+12
Lagos · Nigeria	Africa/Lagos	UTC+1

What is the time difference between Auckland and Lagos?: Auckland is 11h ahead of Lagos. This gap is not fixed: daylight saving time moves it between +11h and +12h during the year.
Does the time difference between Auckland and Lagos change during the year?: Yes. Because of daylight saving time the gap moves between +11h and +12h.
When is the best time to call Lagos from Auckland?: There is no overlap between standard working days, so one side has to call early or late.