The study looked at a total of 216 child contact applications - known as child arrangement orders - which are usually between private individuals.
It was found that, of the cases with domestic abuse allegations, 89 per cent also involved other safeguarding concerns such as substance abuse or mental health problems.
Researchers found that in 23 per cent of cases involving domestic abuse allegations, unsupervised contact was ordered at the first hearing. Some sort of contact, whether supervised or unsupervised, was ordered in 44 per cent of cases at the first hearing.
The report states that in the majority of cases featuring domestic abuse allegations where unsupervised contact was ordered, such contact had been taking place prior to the application to the court.
"The sample cases provided a complex picture of domestic abuse within family proceedings and it was uncommon for domestic abuse allegations to feature in isolation from other safeguarding concerns," the report states.
"This demonstrates the substantial challenge for courts in determining which cases can safely proceed to contact with the child."
The study's results are set against a backdrop of significant rises in the number of children affected by domestic violence.
In May child protection charity Buttle UK published figures showing it received more than 9,000 referrals in 2016 about children affected by domestic abuse, an increase of just under a third on 2015.
The Women's Aid and Cafcass research also found children had strong views about contact with the abusive parent and fathers were found to be three times more likely the subject of domestic abuse allegations than mothers.
Among the study's key findings were that family court advisers recommended either indirect or no contact in more than a quarter of cases where domestic abuse was alleged.
Anthony Douglas, chief executive of Cafcass said: "This research highlights the complexity for the courts faced with making a safe decision for children against a backdrop of disputed allegations and multiple safeguarding concerns.
"We have looked into those cases where unsupervised contact was ordered, as a further assurance to our work carried out during proceedings: we were satisfied that in each case action had been taken to manage risk relating to domestic abuse."
Katie Ghose, chief executive of Women's Aid, said a report the organisation published last year, called 19 Child Homicides, told the stories of children intentionally killed by a parent who was also a known perpetrator of domestic abuse.
"These killings were made possible through unsafe child contact arrangements, formal and informal; and most worryingly, over half of these arrangements were ordered through the courts.
"The research carried out with Cafcass helps build a picture of what is needed to ensure the future safety of women and children survivors of domestic abuse."
In the Queen's Speech the government set out plans to establish a domestic violence and abuse commissioner and define domestic abuse in law.
It also outlined measures to ensure that if abusive behaviour involves a child, a court can impose a sentence that reflects the life-long impact that abuse can have.