From the Open Europe "survey":

Ms Shankar added: ‘There is overwhelming support for migrants coming to the UK in areas where we have skills shortages and to do socially-useful roles, for example as doctors, nurses or teachers.'

At the same time, the public believe that migrants’ access to welfare and public services should be restricted.

Ah, the good old class divide feeds into to racism, as ever. To reinterpret the rather benign-sounding take quoted above:

"Skilled workers (of the kind we fail to train ourselves) can come here because they are useful for filling our gaps (and, by the way, to hell with their native countries who will just have to do without them). At the same time, though, they mustn't expect to be able to have the same access to our schools, doctors and hospitals that we native Brits get, however useful these people are, and we definitely don't want these foreigners to have access to welfare."

This is the aspiration towards exploiting foreigners, especially the "useful" ones, which has tainted the history of this country for hundreds of years. We will have them under sufferance but they mustn't expect equality while they're here. And the referendum leave campaign relentlessly emphasised the "lack of control of our borders" and the need for some kind of points system (a nice way of saying that we won't take foreign riffraff). It's completely disingenuous to suggest that that didn't exploit the most base racist instincts of many people. In fact, given the closeness of the vote, it wouldn't be a stretch to say that it might just have swung the result.