You don't give a timeframe for the family in question. But given the name "Napoleon" among the children, it would appear to be mid-1800s to about 1880, which is when the name was popular in Canada.
I can't find them in the 1870 State census index, though of course they might have moved on by then.
For what it's worth, the spelling "Carrow" is not the original French spelling. It may be part of your problem in tracing the family. You're probably looking for the name "Caron", "Carron" or "Charron" (which is also often seen as "Ducharme"...look up the term "dit" to understand how bizarre this is).
Did a statewide search of the 1870 census on the name Prudhomme and only had one hit...Leone in Detroit.
If you can give us a timeframe when you knew they were in Bay City, it might help more easily pinpoint them. There are undoubtedly records in one of the Catholic parishes in town, but until we know when they hit town, it's only a guess about which one.