A longstanding culture of disregard for victims of rape, sexual assaults and other major crime was exhibited by the police force at the centre of the Rotherham child abuse scandal, the official watchdog concluded on Thursday.

As its embattled police commissioner, Shaun Wright, refused demands to quit from David Cameron, an inspection of South Yorkshire police found that officers spent a "great deal of time trying to disprove" the victims of sexual abuse rather than investigate their complaints.

The damning report by Her Majesty's Inspectorate of Constabulary (HMIC) also found that the force should have pursued a fifth of the violent offences, including rapes, that it wrongly dismissed as "no crime" between November 2012 and October 2013.

Labour MP John Mann told the Guardian that the government should take over South Yorkshire police if Wright continued to refuse to resign. He said: "Shaun Wright is a tiny part of the problem, but he needs to resign immediately. He has no local support whatsoever.

"We're getting to the stage where the government should take over South Yorkshire police. It would be unprecedented, but it's an inevitable consequence if Shaun Wright doesn't resign."

Wright, who was responsible for Rotherham children's services from 2005 to 2010, was clinging on to his job despite calls for him to go from Cameron, the deputy prime minister, Nick Clegg, and the home secretary, Theresa May. He resigned from the Labour party on Wednesday night having been faced with suspension, but refused to leave his £85,000-a-year post.

Cameron said that the "right decision" for Wright would be "to resign and take full responsibility for what happened". Instead it fell to Wright's deputy, Tracey Cheetham, to stand down, also calling on him to follow suit. She said: "It is vital for people to have confidence in the office of police and crime commissioner and, with this in mind, I believe it would have been the right thing for Shaun Wright to resign."

The police force and Rotherham council have come under pressure this week after the publication of a damning report by Prof Alexis Jay, which concluded that there had been 1,400 cases of child sexual exploitation in the Yorkshire town between 1997 and 2013 – caused in part by "blatant" failures by both agencies.

Tom Winsor, the HMIC's chief inspector, said that South Yorkshire police was still failing to record crimes against children properly. He ordered the force to end immediately its culture of "investigate-to-record", where officers do not record incidents as possible crimes until they have been investigated.

He said: "This culture of dealing with reports of crime shows a disregard for victims and is unacceptable; it hides the true extent of the picture of crime from the force and is particularly concerning when the offences investigated by this unit are often of the most serious nature and victims are often the most vulnerable."

Inspectors examined 53 reports to South Yorkshire's specialist departments, the report said. Out of those, 34 crimes should have been recorded – but only 18 were. "This level of under-recorded crime is a significant cause of concern and is a matter of material and urgent importance, particularly as some of these relate to violence and sexual assault against vulnerable children," Winsor said.

The force's chief constable, David Crompton, was also warned by a second watchdog that a number of his senior serving officers could face disciplinary action or dismissal. In a letter to Crompton, the Independent Police Complaints Commission warned of "serious and troubling concerns about the action or inaction of South Yorkshire police".

Meanwhile, the education secretary, Nicky Morgan, ordered an early Ofsted inspection of child protection services in Rotherham. The children's minister, Edward Timpson, also wrote a strongly-worded letter to Rotherham council leaders warning them to expect an early visit from Ofsted inspectors scrutinising its child protection and looked after children's services. Morgan separately warned the council: "I will not hesitate to take further action if necessary to ensure children are safe."

South Yorkshire police said it is currently investigating 173 cases of suspected child sexual exploitation – including 32 in Rotherham alone. A string of former council leaders apologised for failing to stop the abuse of 1,400 vulnerable children over 16 years. Joyce Thacker, the current head of children's services at Rotherham council, said she was "appalled" at revelations about abuse in the town – but vowed not to quit and said protection services had been transformed since 2010.

South Yorkshire police said it would inform the IPCC if there was any evidence that officers failed to properly investigate child sexual exploitation or had covered up evidence. The force said it welcomed the HMIC's findings and was actively working to implement the recommendations made in the report.

A spokeswoman said: "It is important to recognise that this report is about the recording of crime and not how we investigate crime. The force's public protection unit is victim-led and officers take great care to ensure all victims of this type of crime are fully supported throughout an investigation, from the moment a report is received."The HMIC verdict echoed the finding by Prof Alexis Jay that the police "gave no priority to child sexual exploitation, regarding many child victims with contempt and failing to act on their abuse as a crime". In her excoriating report, Jay detailed how children as young as 11 were deemed by police to be having consensual sexual intercourse when in fact they were being raped and abused by adults. Their rapists were, at most, let off with police cautions, free to continue their abuse and to threaten victims' families.