I think you've got it a bit backwards, OP, though the impact is the same.
It's not that the cuts are actually targeted so much at the north as a whole (though there has definitely been some nakedly political shifting of funds). It's as much that some places are less prosperous already, with less private sector employment and therefore more reliant on public sector employment, so when public sector cuts happen the impact on the local economy is much greater. And so the shops and restaurants close, etc.
Until we tackle our woeful productivity performance outside London and pockets of the SE (by no means all of it) the pattern will continue. I saw a horrifying study recently that looked at productivity in European cities - while London's per worker output is near the top, all of our other big cities' productivity are way below average. That then hits satellite towns and rural areas.
That's not saying the people in Leeds, Sheffield, Bristol, Newscastle etc don't work hard. They do. But there is a huge and chronic lack of business and industrial investment in those places that would raise output. And that's not just about straight government spending, it's about our total lack of any kind of industrial strategy, proper long term plans to support and build strategically important industries (look at the semi-conductor issue and how the Dutch had a long term strategy, and saw it through even though they were losing money in the short term, now they're raking in the cash, employing huge numbers, exporting everywhere and paying lots of tax), proper investment in skills and automation. And yes there are spending elements to that but much more important is how that spending is targeted and sticking to a long term plan.
We are at the natural result of a dedication to the wisdom of markets and free market economics. Markets turn out to be dreadful at building and supporting thriving communities. They're not always right. But unfortunately nearly 30 of the last 43 years, we've been run by people who have blind faith and devotion to markets over economic planning, and its led to concentrations of productivity and prosperity, and screw the rest.