Maybe a newspaper for the farming industry? There's one in the UK called 'Farmer's Weekly'. That could be called a farm paper. But we'd need the full context to be sure.

I have never heard of a farm paper could it be a misspelling of term paper?

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It simply means a publication for the agriculture industry. I live on a farm in Canada and a popular farm paper here is ‘The Western Producer’. It contains articles and advertising pertaining to all areas of agriculture.

I suspect this is not very recent. When US agriculture was comprised of mostly family farms, these communities might have a weekly farm news publication; which publication addressed issues affecting farmers. However, as large corporations took over from the smaller farms, many of these publications were lost and/or absorbed into a farming section in the more general news media. I still see them, occasionally, when I’m in a rural farming community - but not often. I would say they are very regional.