It would be fair to say that Liam Byrne has not been the most popular member of the Shadow Cabinet. I sat down with him in the Arc Café as the scandal that made him a household name hit the news again.

As outgoing Chief Secretary to the Treasury in 2010, he left a note to his successor which read: "Dear Chief Secretary, I'm afraid there is no money. Kind regards – and good luck!" That note was brought up time and time again by the governing coalition to support its claims of Labour's profligacy in office. I almost regret having to ask about this, but he shrugs it off.

"The truth is, in politics there is dark humour. I was honouring an old tradition." He really is all politician, sporting a socks and a tie in bright Labour red and – just visible beneath his jacket sleeves – Union Jack cufflinks. True to form, he immediately begins talking about the other things he claims to have "left", such as a triple-A credit rating.

Did he expect David Laws, the next Chief Secretary, to leak the note?

"No, there's an unwritten tradition amongst MPs that you never disclose private correspondence." He gives me the answer he has given several other publications, referring to the expenses scandal that abruptly halted the career of among the most talented Liberal Democrat front-benchers: "David Laws is someone who was hauled before Parliament for some unfortunate expenses claims a couple of weeks after, and slammed the recovery into slowdown. People will forgive me for saying I won't take lessons in ethics or economics from David Laws."

That was some time ago, now, and his current brief is the Shadow Minister for Universities, Science and Skills. This puts him opposite Greg Clark and Nick Boles, both of the Conservative Party. The Labour policy on tuition fees has been criticised as being, counter-intuitively, regressive: the Institute for Fiscal Studies said that the primary beneficiaries of a cut in tuition fees to £6,000 per annum would be mid- to high-income graduates.

"What that criticism ignores is the fact that the current system is going bust. No change is not an option," says Byrne briskly. "The reason it's a progressive plan is that, in order to pay for it, we're asking those who are richest to lose a few of their pension tax perks. The very richest get a 45% tax break on their pensions, and we're bringing that down to the same level as any other basic rate taxpayer. We'd cap pension pots at about £1m. That's a very progressive way of stopping the system going bust."

It goes beyond tuition fees, he says. "One of the things we've heard from students all over the country is that change is needed in terms not only of the headline fees but also increasing maintenance grants. [...] We are determined to raise maintenance: maintenance grants go up by £400 for over half of students, so we'll radically extend eligibility."

Is it affordable to have half the population in higher education, as was, and remains, Labour's target? 

"We can't afford not to," Byrne fires back. " What we've got to do is to make some changes to make it possible for even more people to go to higher education in the future.

"Globally, the only way that Britain stays competitive is to get as many people to degree level as possible because ultimately as a country unless we grow smarter we're going to grow poorer."

Speaking of poverty, his time as the shadow Work and Pensions Secretary, opposite Iain Duncan Smith, let to criticism that Labour was not working hard enough to differentiate itself from the Conservatives. His departure from the position in the Shadow Cabinet reshuffle in September 2013 was quiet and widely viewed as having been inevitable. Has Labour changed since 2010 on welfare – or, as Ed Miliband insists on calling it, "social security"?

"Labour's a party that believes Britain does well if we're a country of ambition and compassion. A strong welfare state is important to us because that's the way"—he stops to correct himself—"one of the better ways we've invented of looking after each other. And that compassionate line runs right the way through us.

"We do, as a party, think social security needs reforming and modernising for the 21st century. That's one of the reasons we've said the bedroom tax would have to go, that's a big issue here in Cambridge [...] but at the end of the day the best way to support people getting on in life is a job." To that end, he says, Labour would reintroduce the Future Jobs Fund, paid for by a tax on bankers' bonuses.

And what about migration, another issue on which Labour takes quite a lot of criticism for not being sufficiently un-Tory? Long ago, in 2006, Byrne was appointed as Borders and Immigration Minister; his successor, Phil Woolas, had his re-election in 2010 invalidated by an electoral court for falsely linking his Liberal Democrat opponent to Islamic extremism. Can Labour be trusted not to cut off the immigration that props up university finances?

"Absolutely," he says quickly. "I introduced the points system as Immigration Minister, and I went to incredible care to make sure that it worked for students, and it worked for universities. We did that in two ways: one is we didn't have a net migration target for students, and we had a brilliant post-study work visa that was crucial for universities recruiting foreign students. Now, we've had the first fall in international students for 29 years because of the ham-fisted approach this govt's taken to student visas, and the Lib Dems haven't stopped that [...] One of the things we will do is take students back out of the net migration target and reintroduce the post-study work visa."

I ask him if that's the same policy as UKIP is running on: this is a little unfair, as UKIP has no clear position on special post-study work visas, but does advocate for counting students separately from ordinary immigration.

"The UKIP position on immigration has now changed so often it's hard to pin down what it really is. Nigel Farage says one thing, the party spokesman says another. You're probably a better interpreter than me!" he jokes.

With that our time is up. I bid goodbye to Liam Byrne, and to Daniel Zeichner and the small phalanx of CULC members campaigning at the Sidgwick Site. In a seat where Labour's chances, especially with students, are vulnerable to the hard-left allure of the Green Party, one does have to wonder at the wisdom of sending one of Labour's longest-standing centrists to help a candidate running on a strong social-justice campaign. But, in the end, it probably won't matter: all five parties, even UKIP, are competing for students' affections. Pulled in so many different ways, the discerning student is spoilt for choice.

@rtrnicholl