A beck is a North of England dialect term for a small stream through a Gill or valley.
A beck flows past The Dog's Home and is crossed by stepping stones. It goes through a pipe (culvert) under the road and then through our coppice to the river. Heavy rail makes the beck overflow and the water runs down through the trees (PM4). Hence perhaps the name Beckfoot.
A stream flows into Horseshoe Cove, which is sometimes called a beck; Titty says when she and Roger are exploring: We’ll get back to the beck at once and We can’t get lost if we keep close to the beck (SD4). When they go to sleep after the shipwreck, there is the noise of the beck hurrying past the tent doors and the noise of the beck hurrying through the middle of the camp (SD10,11).
In the fog on the moor Titty and Roger find the "beck" or "little stream" and think the compass is wrong because the beck flows west not east. But they realise when the fog lifts that the stream was going the wrong way .... and they must have turned right round in the fog (SD28,29). After Roger twists his ankle, Titty and Young Billy go back to Roger who is Top of the wood. By the beckside (SD30).
When Captain Flint asked the Ds where they got water from (as the lake is frozen) Dot says We bring it from the stream - the beck I mean (WH21).
When Titty mentions the stream near the Hebridean Pict-house, Dorothea says Burn .... not stream and Roger says It would be a beck if we were at Holly Howe (GN4).