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My K2 will run from a jerry can which makes it easier to keep the hose in place than with a bucket.
I use a couple of metres of garden hose with a hozelock on the end - the 'official' Karcher hoze has a filter for use with water butts and other dubious sources, but if you're filling the bucket from a tap then particulates shouldn't be a problem.
Expect lower pressure at the nozzle than when attached to the mains - not quite enough for the patio nozzle to work properly, good enough for fresh mud on the bikes / cars / caravan.
not really a DIY question I suppose but not sure where to put it - can you run karcher pressure washers (e.g. the K2) from a supply of water in a large bucket (i.e. instead of connecting it to a supply of water from a tap)? I have an awkward kitchen tap and don't think I'd be able to mount the hose to it