Rogue potatoes - should I encourage them?

I took over a lovely, clear, allotment last November and covered it with weed-supressing membrane. Have been visiting regularly to start planting this spring. Today noticed several humps in the membrane and peered underneath - it's spuds! Dug one up - it's a Desiree or other red variety - just a withered, slug-nibbled lump. The plants are a bit bent over and the leaves are small - but they are vigorous. Question is - are they likely to go on to become good croppers, or are they just the last gasp of last year's harvest? There's absolutely nothing else growing under the covering that's been down since last November, so I feel a bit sorry for these plucky tubers . . .
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I always thought that volunteer potatoes were likely to harbour diseases, but that might just be an old husband's tale.
If you live in Derbyshire, as I do.
Dig it out, you then start with a clean slate rather than having the odd spud growing randomly in veg beds which may not harvest at the same time as spuds and disturb their roots when you dig up the spuds.
Free potatoes, half the work!
Thanks for the prompt replies; seems the jury's out on this one. I did have potatoes pencilled in for one of the areas, then decided not to bother. I might just leave one or two in that area and get rid of the rest.