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As far as I can remember:
the price of 'sugar' in the EU is above the World market price for two reasons.Firms with Colonial/Slave era histories like Tate & Lyle can strong arm small producers in the caribbean, (and possibly elsewhere) to keep the sugar cane growers on the breadline, and keep all the added value for themselves.
France, East Anglia, (and potentially other tracts of agricultural land within the EU), can produce beet-based sugar. It provides employment and a food-security, local source of low-carbon/low food miles sugar.
The EU also gives special tariff arrangements to formerly colonially impoverished states in the Caribbean as long as the production meets some EU standards, 'Organic', 'Fairtrade', etc.
[I seem to remember the French were better at this than the UK, as Tate & Lyle have always been politically connected, long before David 'thick as mince' Davis appeared on the English political scene].This reduces the likelihood of impoverished Caribbean farmers reverting to the guaranteed crop of choice:- weed.
The sugar Brexit thing is about the raw materials they use to extract their sugar from. It can come from either cane or beets, one of which gets an EU subsidy to grow in the UK (beets? I think...)
It's something along the lines of Tate and Lyle using cane, which is imported. So their competitors no longer get to use subsidised raw materials, hence their pro Brexit stance.
Also, David David worked there a long time, so there's that.