If farmers simply stopped their vehicles before they entered the public roads (and there is normally a verge or a gateway to do so) and cleaned the great clumps of mud off, they wouldn't actually need to clean the roads! Common sense really.
As far as horse poo is concerned there is no legislation so I think you are probably stuck with it shaveoff. Although personally, I live not far from Newmarket, and on the ocassions I have visted the town, I have never seen it as a particular problem.