It does sound a bit like "chumi", doesn't it? https://translate.google.com/#view=home&op=translate&sl=auto&tl=nl&text=Dzie%C4%87mi and click the speaker symbol does sound rather better, IMHO.
Probably worth reporting, if you get it again.
"Children" is a plural of "child" and you shouldn't put indefinite article in front of plurals in English – think of it like that: if you aren't talking about specific group of nouns(and in such a situation, you would use the definite 'the'), a plural denotes an indefinite mass of all the nouns you name with the plural, so you don't need to also add the indefinite article – the plural takes care of it on its own. ;)
Interesting question! I guess that perhaps "dziecko" is a little bit irregular in its declension, but there may be some rule which explains the details. Its full declension can be seen here https://en.wiktionary.org/wiki/dziecko and click on the "show declension" tab.
The plural forms of "dziecko" end in "ci", except where the "mi" is added in the Instrumental case; I'm going to suggest that because the "i" of the normal "ci" ending is suppressed by the "mi", that the "ci" becomes "ć".
I've seen similar in other words. There is the city of "Poznań" but "in Poznań" is translated as "w Poznaniu".
My guess would be that in "dzieci" the c is followed by an i, which is already a very soft sound.
In "dziećmi" the c is followed by an m, which is definetly not soft, so the c turns into a ć, which makes the consonant softer. But just my thoughts, might not be the real reason :)
I think you may have misunderstood; we generally, but probably incorrectly, speak of "ć" of being a "c" with an accent. We also refer to what you are calling the "accent" as the "stress", so we'd say that "one should put the stress on "dzie" ". Nothing to do with the sound quality in this case :-) .