Immigration is one of those cases where the evidence seems to be almost all on one side, and perception very largely on the other. Without immigrants, the NHS (and plenty of other services and industries, not least in agriculture) would fall over tomorrow. Immigrants more than pay their way in taxes: they're an economic good. Where immigration has led to problems (for example in Lincolnshire) it's because immigrants are perceived to be taking houses, school places, hospital beds, and other things from people who already live here. But, since it's the job of government to provide those things (with the taxes that immigrants among others pay), shouldn't the government rather than immigrants be getting the blame?
That is the burden of Corbyn's song, and it seems a no-brainer to me. However, while many people in Labour would agree on the facts, they argue that the perception matters more. These people believe that the general population is too stupid, or too racist, to be persuaded - that they are in fact intractable in their opposition to immigration and that, this being the case, the Labour party must set out its stall against immigration too, as a sop to the ignorance and prejudice of its core demographic.
This is called realism - or triangulation, if you prefer. To me it seems a hugely patronising attitude. It also seems like appeasement, if not of racism then of ignorance, but more likely both. It's a doomed strategy too, because no matter how "tough" Labour may be on immigrants UKIP will always be tougher: there's no winning on that wicket. It's lazy, because it foregoes the real job of trying to persuade people in favour of making "the right noises." It's fundamentally dishonest, because it involves blaming immigrants for a situation that these people know (and privately acknowledge) is the result of government policy. Finally, it's just plain wicked, because no matter how anodyne a packaging you wrap it in, it's going to sow mistrust and almost certainly abuse and violence too (those things the right wing of Labour has spent all summer decrying) - all in the hope of swiping a few, halfhearted votes. And the people who advocate it know this.