I don't think there is any difference in the way you would pronounce "man~u~fact~u~rer" or "ma-nu-fac-tu-rer" when you say them aloud. This is mostly an issue when writing. It is a visual, publication rule. It comes into play when you need to break a word across a line and need to know where to put the hyphen. And it is only important in the most careful, formal writing. If you are writing a letter to a friend, or even a paper in a classroom, nobody is going to notice whether you write manufac- turer or manufact- urer. Nowadays, in case you haven't noticed, in addition to built-in spelling dictionaries, word processors also have built-in hyphenation dictionaries. They know where to hyphenate words if they need to. You can find rules, but none of them are reliable. When in doubt, do what native English speakers do when we are not sure how to break a word into syllables: look up the word in a dictionary. Or use a word processor and trust its automatic hyphenation feature. By the way, according to the first dictionary I checked, the accepted syllable breaks in "manufacturer" are "man·u·fac·tur·er."

Excellent question, Julia. The rules of dividing words into syllables are different depending on whether you’re referring to orthography or pronunciation. Orthographically, I recommend never dividing words — there’s no reason to (as Dan explains). Referring to pronunciation, native speakers are of course happily unaware of the “rules”, but they follow them unconsciously. This affects rhythm, stress, and vowel timbre in a very subtle way. Getting to your specific question, the N in “manufacturer” must go with the first syllable, because an English syllable cannot end in the vowel /æ/. Phonotactically, the only vowels that can end a syllable in English are the tense vowels and the schwa. All the lax vowels occur only in closed syllables. You can observe for yourself that no word ends in a lax vowel. The same is true of syllables within a word.

There are definite rules for dividing words into syllables when writing English. Here they are: https://www.logicofenglish.com/blog/65-syllables/285-how-to-divide-a-word-into-syllables

Long words don't really split when you say them. Perhaps you are wondering where to put the stress? In any event, it might be worth checking significant new words in an online dictionary which has sound samples - many of them do. https://www.teachingenglish.org.uk/article/word-stress