I interpreted it 2 ways:— 1)The unhappy condition of the place where
the child live (as a member of the family or household) can affect his
behaviour. 2)The unhappy condition or surrounding in which the home
exist can affect his behaviour.
The first interpretation is more correct. But, "The unhappy condition of
the child's life when the child is at home as a member of his family and household can affect his behaviour" would be better.
A "home environment" (especially when used together with the word "unhappy") is less about the physical place where that the child goes home to, and is more about a combination of (1) the child's family and household interpersonal relationships, (2) the emotional impact of the physical conditions in the place where the child's household lives (think of Harry Potter forced to live in a cupboard under the stairs, or a child who is chained to her bed at night), and (3) the neighborhood in which the place where the child's household lives in embedded, in approximately that order of importance.
If someone says that a child has an "unhappy home environment" they are more likely to be referring to high levels of interpersonal conflict between members of the household and to the child being treated poorly as a person by other household members (perhaps even to the level of child abuse or domestic violence), rather than to the house where that household lives being shabby.
The neighborhood that the house where that household lives is primarily relevant to an "unhappy home environment" through the emotional impact it has on members of the household. For example, a neighborhood would contribute to an unhappy home environment if the neighborhood is one where the child lives in constant fear of violence, or doesn't fit in because the child is a minority of some kind in the neighborhood.