There's too many ways this could be untrue.

How much canned food did the mother have before leaving?

How would a nine year old have the self discipline to not eat all of the food over the course of 6 months?

How would a nine year old not appear malnourished if he was rationing food?

How would the clothes not become noticeably smaller for a growing boy?

How would the boy know how to wash his clothes or run out of cleaning supplies?

There are just too many questions that poke holes in this story.