The object trailing from the cable to the left is the windscreen washer pump motor. There's another inside the engine bay that runs the headlamp washer, along with another cable that connects to the washer tank level sensor. I then checked over the tank itself.
I figured something must be wrong when no liquid started pouring out of the holes in the bottom. Pulling the filters out revealed: this!
Ewww.
A bit of washing up liquid and a brush later, back to factory finish. The left one is the headlight washer filter, the right one is the windscreen washer filter.
I gave the tank a bit of a clean and removed the old washer fluid as well (it was a bit sludgy in there as well).
That is cleaner!
I then replaced everything, tightened the 10mm bolt that holds it all in and...
Yay! Result. Estimated cost at garage - £40. Actual Cost 0.0.