You have to be careful when reading these types of statistics. Here:
The UK has six of the poorest 10 regions comparing GDP per person against nearby countries, but this doesn’t mean the people who live there are poorer.
fullfact.org
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There have always been large inequalities in Britain, Dickens & Engels amongst others highlighted the plight of the poor 100+ years ago. The Nazis had detailed plans of British slums to use for propaganda purposes.
It got better after the war, and then widened again during the Thatcher years. Being in the EU hasn't done much to alleviate it.
The problems are structural, endemic in the British economic/political system and you could argue being in the single market has made it worse due to the geographic proximity of London and the South East to the continent. It can be quicker to get a train from London to Paris than from London to Manchester for example.
You cannot, in my opinion redress these imbalaces whilst staying within the EU.