Tens of thousands of students and school pupils walked out of class, marched, and occupied buildings around the country in the second day of mass action within a fortnight to protest at education cuts and higher tuition fees.

Amid more than a dozen protests, estimated by some to involve up to 130,000 students, there were isolated incidents of violence and skirmishes with police, mostly in central London.

The police tactic of penning students into a so-called kettle near Parliament Square for several hours caused anger, but appeared to contain the disorder.

One exception came as night fell, when police mounted on horses charged at about 1,000 students gathered near Trafalgar Square. The protesters ran through the area, with officers following. Students then hurled chairs and traffic cones into the road as bemused tourists looked on. At least two bus windows were smashed and shops were also attacked.

The coalition government condemned the protests, saying they were being hijacked by extremist groups. The education secretary, Michael Gove, gave a notably combative response, urging the media not to give the violent minority "the oxygen of publicity", a resonant phrase associated closely with Margaret Thatcher's efforts in the 1980s to deny the IRA television coverage.

Gove said the government would not waver, adding: "I respond to arguments, I do not respond to violence."

In contrast, Nick Clegg, the deputy prime minister, whose pre-election pledge to oppose increased tuition fees has made him the focus of student anger, spoke of his "massive regret" in having to rescind the promise.

"I regret of course that I can't keep the promise that I made because – just as in life – sometimes you are not fully in control of all the things you need to deliver those pledges," he told one of several angry callers to BBC Radio 2's Jeremy Vine Show. "Of course I massively regret finding myself in this situation."

But he said that the fact the Liberal Democrats had been forced into a coalition, and that the country's finances were worse than they had anticipated, meant they had to accept "compromise".

Asked about his reaction to footage, earlier in the week, of students hanging him in effigy, Clegg said: "I'm developing a thick skin."

In a further sign of the developing pressure on the government's cuts programme, Len McCluskey, the new leader of Unite, Britain's biggest trade union, put himself and his union at the forefront of "an alliance of resistance". In an interview in the Guardian, McCluskey says: "There is an anger building up the likes of which we have not seen in our country since the poll tax."

The biggest single protest was in London, where about 5,000 people – many of them noticeably younger than those who took part in the previous mass protest on 10 November – spent hours kettled" in Whitehall as officers sought to prevent a repeat of the chaotic scenes when protesters burst through police lines to storm the Conservative party headquarters. Thousands more marched elsewhere around the country while others staged sit-ins at university buildings.

About 3,000 higher education students and school pupils gathered to protest in central Manchester, where there were four arrests, and a similar number gathered in Liverpool. A crowd of around 2,000 people protested in Sheffield, with about 1,000 doing so in Leeds and 3,000 in Brighton. There were scuffles in Cambridge as crowds attempted to storm the university's Senate House.

A total of 17 people were treated for injuries in London. Of them, 13 needed hospital treatment, including two police officers, one with a broken arm. Police said 32 people had been arrested. One 19-year-old art student was pictured trying to stop masked marchers attacking the van. "We're going to be portrayed badly in the media," she shouted at them. "We're just wrecking a police van."

After being forced to apologise for the mayhem two weeks ago when fewer than 250 police were unable to marshal a crowd of more than 50,000, Scotland Yard sent almost four times as many officers onto the streets and quickly penned marchers into a section of streets. Late last night some parents arrived at the police cordon pleading for their children to be released. The worst violence erupted after 6pm as officers let the marchers leave.