No. It's wrong. Anyone who told you that it means "some of the" is wrong!
"Du" is the partitive article, and mean an indefinite quantity, it only mean "some" and NOT "some of the". Your mistake is to translate it word by word.
Only translate the partitive articles with "some" or no article in English. When you have a partitive article in two words, remember that "de la" is like a single word, it's a whole expression, and the "de" can't be translated with "of" literally here!
See here for Partitive articles:
Is there any way of saying some of the in a sentence like this, or would it just not be said? In English, if there were a table laden with food - we might say I'll just have some sandwiches (the ones on the table is implied) or I'll just have some of the sandwiches It would mean the same thing because of the context.
I think the difficulty for many English native speakers learning french is the "du" can in some cases be translated as "some" as in "du pain" but in other cases it translates as "of the" such as "le chapeau du garçon".
It is then easy forget that these are two completely different meanings - the result the two meanings are run together "some" + "of the" = "some of the". Of course this is wrong as you have explained but it is a very common confusion among english speakers learning french.
I did a similar mistake, then I realized that is the subject of the sentence were in the plural the article should be "Les enfants mangent du pain".
The trick on these cases you need to detect if it is plural or singular is to watch for the articles and extra information since there is no audible difference in the verbal terminations.
I actually thought that this said, le femme mange du pomme- making me entirely wrong. I am struggling with differentiating all these word when so many letters seem silent or muted in french- does anybody have any tips or suggestions for more practice with both pronouncing and understanding better?
It is clear, it just takes practice - PERCE_NEIGE talks about it a bit earlier on this thread.
You can go to Forvo and key in the words separately or use goggle translate which is quite good for this particular pair.
Key the woman and the child (as a phrase) into google translate with French as the destination language. It will say la femme et l'enfant in the box to the right and there is a little mic at the top. You can listen to this to train your ear. It will come with practise.