This phrase occurs in 2nd paragraph of 东南亚民间故事。上，85页（福建人民出版社出版（福州得贵巷２７号）１９８２年５月第一版，１９８２年５月第一次印刷) under the title 蛇王子与三姐妹, the whole paragraph reads as follows:
每天，寡妇到集上去。她必须经过一棵芒果树，树上结满小而甜的芒果。寡妇每次路过总要留意树下有无夜间从枝头掉下的芒果。
Some users might think (不分)昼夜 instead of 有无夜间 might make better sense. There are other versions of this Mianmar folktale, involving a mango or fig (无花果) tree but users have not yet found another one where the widow must be careful about the mangoes (figs) falling from the tree.
The 有无 ("to have or have not") here is similar to 有没有, but it pertains to the existence of something, in this case, the existence of certain mangoes:
……有无芒果。
...whether or not there's mangoes.……有无（夜间从枝头掉下）的芒果。
...whether or not there's mangoes falling of branches at night.寡妇每次路过总要留意树下有无夜间从枝头掉下的芒果。
Every time the widow passes by, she pays attention to whether or not there were any mangoes under the tree which had fallen of the branches during the night.
It's not part of a phrase 有无夜间, and it's not related to 不分昼夜 nor 无花果. The phrase 夜间从枝头掉下 functions as an attributive (定语) modifying 芒果.
Schöne Grüße alter Freund, user6065!
有无: in case
结满: covered with
每天，寡妇到集上去。
Every day the widow goes to the market.
她必须经过一棵芒果树，树上结满小而甜的芒果。
She has to pass by a mango tree full of small, sweet mangoes.
寡妇每次路过总要留意树下有无夜间从枝头掉下的芒果。
Nevertheless, every time the widow passes by (the mango tree), she always pays attention, in case any mangoes have fallen from the branches in the night.
有无夜间 ✘ 有无/夜间/从枝头掉下的/芒果 ✔ '有无夜间' is not a word. 有无 is equal to '有没有' in this sentence, always be used for whether something exist or behaviors execute. For example, boy A asks boy B if B had a ruler, A could says:'有无尺子' or '尺子有无'(Although '有无' is more common in Chinese ancient literatures and rarely said in dailylife). So the sentence can be '寡妇/每次/路过/总要/留意/树下/有没有/夜间/从枝头掉下的/芒果。'. '夜间' means 'at night', '①夜间/②从枝头掉下的/③芒果' means '③mangoes/②falling from branches/①at night'.
The phrase "夜间(during the night)" merely to create the image that the widow goes to the market very early in the morning (清晨) every day, so any mango on the ground would have fallen from the tree over the night.
"(不分)昼夜" does not fit the purpose.
