I believe that the relationship between the type of parent/child relationship and the child's well-being later on in life could be explained by the fact that lower-income families are more likely to be single-parent families. If the mom has to work to support the family, she will not be able to spend as much time and develop a good relationship with the child. Additionally, she will be more drained after a day of work and may not want to play with the child.
I think this is a very good point. Children who have a stay at home parent have many more opportunities to develop a strong bond with that parent. They trust that the parent will be there when they need them, and that dependency can carry over to future relationships. It is also possible that after working all day, the parent's main priority might not be to solidify their relationship with their child. Who has the energy to work, and then come home and spend hours with their energetic child?
That's a pretty particular situation but it is pretty relevant to many lower-income families nowadays. Apart from coming home tired and not wanting to play, relations between the child and mother could be even worse. After a long day at work, the mother might react negatively to the complaining child such as instinctively replying with a raised tone or even harsh looks or movements. This could permanently harm the relationship between the child an caregiver and influence the amount of trust that the child will have in other people in the future.
