According to latest Office for National Statistics figures, 130,000 children in England and Wales live in homes where there is a "high risk" of domestic abuse.
The report cites evidence heard by the committee from charity Action for Children, warning of the destabilising effect domestic abuse has on children's lives.
The charity's evidence details how such abuse can harm the brain and sensory growth of children, their ability to develop healthy relationships as adults, and increases the risk that they will become abusive themselves.
"Children who have experienced domestic abuse often suffer emotional trauma as a result, and this can have a severe, lifelong impact," states the charity's written evidence.
"Infants who are exposed to violence in the home can undergo so much added stress that it can negatively affect brain development and impact on cognitive and sensory growth."
"A quarter of the estimated 130,000 children currently living in households with high-risk domestic abuse are under three years old."
Meanwhile, Barnardo's told the committee that the Domestic Abuse Bill currently "fails to recognise the full picture of risks and vulnerabilities that children affected by domestic abuse experience".
The committee recommends that the effect of domestic abuse on children is recognised in legislation and that ministers develop a strategy for protecting children.
This strategy should include special waiting list status for all NHS services, especially access to child and adolescent mental health services (CAMHS).
Councils should also be required to ensure stability in children's education is taken into account when making decisions around the welfare of families who are forced to flee to a refuge.
Children in a refuge should also be given the same status as looked-after children by being guaranteed a new school place within 20 days, says the committee.
"Children who have experienced domestic abuse risk suffering a range of long-term negative consequences as a result of their experiences and must be able to access the necessary support and health services to help them recover," states the report.
"Children in refuge and other temporary accommodation, and those who have moved home repeatedly to flee domestic abuse, are particularly vulnerable and risk becoming invisible to professionals in the education, health and social care sectors."
The committee also wants to see children given greater protection in court proceedings where domestic abuse is an issue.
Among those to give evidence was Dr Shazia Choudhry, professor of law at Queen Mary, University of London, who is concerned that courts are too often allowing abusive parents to continue having contact with their children.
"Domestic violence policies rarely address the area of private family law, where courts may require ongoing contact between children and abusive parents," she said.
"This is arguably the weakest link in this policy matrix."
The committee is calling on the president of the Family Division to look into how family courts can better protect children who have been affected by domestic abuse.
Judges, magistrates and other professionals involved in child contact cases should also have specialist training on domestic abuse issues. And children should also be better supported in court during such proceedings and given access to specialist children's workers.
"We have particular concerns about the impact on children of court proceedings, and the lack of co-ordinated support for them," states the committee's report.
"There is a need for more specialist children's workers who are trained to recognise the impact of domestic abuse on children, and to ensure that the relevant statutory organisations respond to their needs."
The Domestic Abuse Bill proposes the creation of a domestic abuse commissioner to hold the government to account. The committee wants the commissioner to prioritise a review of how children affected by domestic abuse are treated in court proceedings.
"We recommend that the new commissioner should have, as a priority in the first year of office, to review the impact upon children of the interaction between the family courts, children's services, Cafcass and the police, with particular reference to contact arrangements in domestic violence cases," states the report.
Further concerns are raised by the committee about a lack of funding for refuge places.
According to evidence submitted by Sian Hawkins, head of campaigns and public affairs at Women's Aid, 60 per cent of referrals to refuges are being turned away due to lack of space.
"About 94 children and 90 women every day being turned away from services at their point of need, when they are desperate to flee abuse," Hawkins told the committee.
The flight to emergency accommodation can also be "particularly difficult for families", adds the report.
The End Violence Against Women coalition told MPs that the percentage of women forced to "sofa surf" while seeking a refuge place rose from 39.9 per cent in 2016/17 to 45.8 per cent in 2017/18. More than half of these women had children, added the coalition's evidence.
In addition, less than one in five vacancies posted in 2016/17 on Routes to Support, the national database of domestic abuse support, were able to accommodate a woman with three children.
"Leaving the abuser can also often mean that children have to leave their school and friends, for example, if the refuge place is out of area or if there is a fear that the perpetrator will seek to make contact with the family during drop-off or pick-up at the school," states the report.
"Children may have to change school more than once, as they are moved from one place of refuge or temporary accommodation to another."
Yvette Cooper, chair of the committee, said: "We urgently need more refuge places - provision should be a requirement on local authorities, backed by national ringfenced funding."
Earlier this month, Labour pledged to introduce statutory paid leave for victims of domestic abuse if it wins the next general election.
Shadow minister for women and equalities Dawn Butler said the move could save the lives of more than 100 victims a year as it provides them with the chance to find help without putting their employment at risk.
Publication of the home affairs select committee report coincides with a report by the women and equalities select committee into sexual harassment of women and girls in public places.
The report raises concerns that sexual harassment is "almost entirely absent" from a current cross-departmental strategy for tackling violence against women and girls. The committee wants to see this strategy improved to include a plan of action on how public places will be made safe for all women and girls.
"It is not acceptable that women have to change their behaviour to avoid sexual harassment," Maria Miller, chair of the women and equalities select committee, said.
"It has a wider effect on society, contributing to a culture in which sexual violence can be normalised or excused. All of this keeps women and girls unequal."