The council will send its proposed Educational Neglect Policy back to the children and young people's scrutiny committee for a further review.
Councillors say parents have directly raised concerns since the decision was published.
The policy sets out the council's approach to educational neglect - a recognised form of neglect that can damage a child's learning, development and long-term chances.
It explains when schools should act and when the local authority must step in, including commissioning Education Otherwise Than At School (EOTAS) if a school can't provide suitable education.
Section 19 of the Education Act 1996 would apply only after schools have exhausted reasonable adjustments, the council says.
Persistent absence, unsuitable provision and lack of engagement are flagged as strong risk indicators. The council says unclear thresholds have led to inconsistent practice, delayed intervention and, at times, inappropriate reliance on Section 19 or alternative provision.
Opposition councillors say approving the policy through an Individual Executive Member Decision was inappropriate.
The policy will be discussed at the children and young people's scrutiny committee on Thursday (July 30).
Niki Hinman, Local Democracy Reporter
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