If your evaluation of whether a journal is reputable is citation index then the Journal of Marriage and the Family has a very high citation index (1/38 for family studies, 7/137 for sociology). Personally I think citation indices are bollocks so I'd ignore that. I'm not sure how you measure whether a journal is rightwing or not but there is nothing obvious that suggests this particular one is.
The researchers of this article were trying to answer a specific question - do we know if spending overnight time away from the resident parent when a child is very young is likely to harm children's outcomes in the longer term. This is important because very young children are more likely than older children to be living in one parent households and spending time with another caregiver and both parents and sometimes the courts need to consider what arrangements will be in that childs interests. Interestingly the question has not been studied much although plenty of people hold opinions on it.
So they looked at data from a large population based dataset which looked at families with a significant incidence of relationship break-up in the US. And they found there was a small but statistically significant relationship between children under 1 spending more than 1 night a week with a non-resident parent and adjustment problems (read behaviour issues) between 3 and 5. There was also an effect when children aged between 1 and 3 spent more than 1 night a week away from home but this was much much smaller. Interestingly the effect tapered off after age 5 anyway.
Their findings have no application to children who simply do not see their mother at night because she is working or away from home for any other reason. Children who live in a two parent houshold with secure attachments to both resident parents belong to a different demographic to those whose outcomes were followed up in this study.
The kind of residence arrangement they describe is extremely rare in the UK (though common in Australia where shared residence is favoured) so it would be unusual to see children in this situation here. And I would strongly suspect that the findings might be culturally specific. Since the effect is small the best that can be said is that more research is needed.
But is is incredibly unfair on the researchers to accuse them of a rightwing, mother-bashing agenda simply because they wanted to find out what kind of residence arrangements support children's attachments to their caregivers best.