The landmark bill, which received its first reading yesterday (16 July), introduces the first statutory definition of domestic abuse as well as a commissioner to champion victims and survivors.
Charities and councils have continued to call for children to be given more legal protection, and for this to be backed with enough investment in specialist services.
A report last month from a joint committee of MPs and peers had described the draft bill as "silent" on the "plight of children", and called for them to be given the status of victim in affected homes.
A government response to the report this month acknowledged that "more could be done" in the bill, adding that statutory guidance "will outline the range of impact domestic abuse can have on children, as well as appropriate support and referral mechanisms".
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However, charity Action for Children said: "The devil will be in the detail when the bill is published, but we want to see the fundamental changes necessary to provide proper support for children and help them recover from deep and lasting trauma.
"That means putting an end to the current services postcode lottery by guaranteeing specialist support services for children across the country.
"We are committed to working with the next prime minister to seize this opportunity to protect the vulnerable children who are victims of domestic abuse."
Barnado's chief executive Javed Khan added: "The new bill is a step in the right direction, but we would urge the government to go further in recognising the impact of children who are the forgotten victims of domestic abuse.
"Sadly without the right support, children in this situation are at risk of becoming trapped in a life-long cycle of violence.
"We must work together to provide access to vital services so they can recover from harm and work towards a positive future."
Charity SafeLives also highlighted the shortfall in support for young people, claiming that while they experience some of the highest rates of domestic abuse, many are still not able to access specialist support in certain parts of the UK.
The organisation said its recent practitioner survey found that "almost one in five police force areas have no specialist support for young people experiencing domestic abuse".
Chief executive Suzanne Jacob, said: "We trust the government will want to match the bill with the funding required to end domestic abuse for everyone and for good.
"That must include fully funded specialist domestic abuse services in the community for adult and child victims to help families stay safe, and services to challenge perpetrators' behaviour, with a statutory duty on local authorities and financial support from central government."
The Local Government Association urged a greater focus on prevention and early intervention to tackle the root causes.
"The ability of councils to fund services for victims is limited by significant pressures on their budgets, with local authorities increasingly being forced to prioritise spending for those at immediate risk of harm, rather than on vital earlier support services and prevention schemes which help stop domestic abuse occurring in the first place," said councillor Simon Blackburn, chair of the LGA's safer and stronger communities board.
In response to the joint committee report, the government said it had accepted, either partly or fully, most of the recommendations, and had "committed to consider several further" as the legislation progressed through parliament.
"We are keen to ensure that there are sufficient services available to support children affected by domestic abuse," said the response.
"As part of their work on mapping and assessing the provision of services across the country, the domestic abuse commissioner will take into account services available for children," the response states.
Other measures outlined in the bill include new domestic abuse protection notices and domestic abuse protection orders to further protect victims and place restrictions on the actions of offenders.
In addition, victims will no longer be able to be cross-examined by their abusers in the family courts.