Essentially, yes, there is some evidence that being raised in a single-parent home, or a home with a step-parent, is associated with more behavior problems and poorer social adjustment. However, there is only a little bit of evidence for this. And, note that the first finding did not cite diagnosable mental illness, but just more problems, and this was at 4 years old. Other studies are needed for children within a wider age range, and also where psychiatric diagnoses are screened for. One large and good study found more evidence of drug use in children from single parent families. Most of the evidence on problems in single parent homes comes from evidence of effects of parental divorce on childrens' mental health. There is much evidence of greater behavioral problems and more psychiatric diagnoses in children of divorced parents compared to those living with both biological (or adoptive) parents. This evidence is complicated by several issues though, which I will summarize briefly.
It is very important to note that much of the evidence of childrens' mental health problems after a parents' divorce is accounted for by factors other than being now in a single parent home. For example, factors like how much conflict the child was exposed to prior to the divorce, how well the parents get along post-divorce, whether or not the child is allowed to spend time with both parents, the quality of the child's relationships with each parent, whether post-divorce the parent raising the child has to move to a poorer neighborhood, whether the mother (who typically lives with child) has a good support network, whether supportive grandparents are nearby, whether the parents have mental illness (especially maternal depression), how closely the child is monitored for mental health and substance abuse problems after divorce, and even genetic factors related to degree of stress-reactivity etc. all significantly determine how well a child functions. That is, even though many children living with a divorced parent do experience mental health problems, many do not. What determines the mental health of the child seems to be more about changes in lifestyle and the level of stress pre- and post-divorce, and even genetics, and not living with just one parent per se.
So, to summarize, on average, children in single parent homes, especially where there was a divorce, function more poorly than children from dual parent homes. However, mental health problems are not inevitable. Other factors are usually more important in determining the mental health of the child. While the risk for these other factors is higher in divorced families, much can be done to lower the risk or prevent these factors from arising.
Anyway, I hope this answers your question,