Why is "many children don't know their father" wrong? Is this an issue in English? Because every child has only one father.
It's because "mnoho" is an indefinite numeral. Czech does not view numerals higher than 4 (ie. 5+ and including the indefinite numerals) individually, but as one group of things/people/etc. That's why for numeral 1 the verb is in the singular, for numerals 2, 3, 4 the verb is in the plural and starting from 5 it is again in the singular, but in this last case, the noun (subject) is in genitive plural! (Jedno dítě nezná..., Dvě děti neznají..., Pět dětí nezná...)
I think their fathers should also be plural to match the plural children
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First you must understand that primary is always the sentence to be translated and the translation comes only after the sentence to be translated is defined. So there is no reason to look what English does when writing an arbitrary Czech sentence.
And the Czech sentence above is a normal natural Czech sentence. That's all.