Something of a guess this, but it fits the facts known to me. I would think that the muck is raw sewage as already stated, referred to in other parts of the country as "night soil" and transferred to its final resting place by a variety of means, which I shall touch on shortly. In this case I would guess the sewage would be collected by horse and cart, taken to a quayside and loaded onto an open barge for conveyance to the Wash, I imagine since this was an essential daily operation which could not be dependant on wind and sail that the barges were originally horse drawn, later replaced by steam power, and possibly later on diesel. The Wash was and still is to an extent extremely polluted and a variety of shellfish like mussels are banned for human consumption. I must be lucky, ate loads of them during rasioning in the 1940s and early 50s, fresh boiled straight off the boats!!
On to the transport of sewage this from the Tinsley/Rotherham/Sheffield area, prior to the giant seage works at Tinsley. The night soil was loaded onto a train in a siding at Ickles, and taken nightly to Toton for disposal in the Trent!!The train was quite notorious (known as the Ickles sh*t train)
It was the fastest scheduled freight train in the timetable, ran 365 days a year; it was the only train in the UK at that time where the guard was allowed to travel on the footplate with the driver and fireman instead of in a brakevan at the rear. Probably would never have survived.
I know that sewage is still transported around the country, on the Terrace at the House of Lords overlooking the Thames, tranquility is spoiled several times an hour by the passing of a barge full of the stuff en route for dumping in the Thames estuary. I am told that Southend on Sea is known locally as Southend on mud, except it isn't mud, think about it!