We have had a number of leaks in a wall which we think are coming from somewhere in the bathroom which is on the other side of the wall. After removing some of the plaster we have found a hole in the wall which is full of black powder, and around the hole is a circle of what looks like rusty coloured sand like powder. Can any one advise on what they think this is please.

Well, that black stuff you describe is most likely mold. And that WILL have to be dealt with as its VERY unhealthy.

Usually by ripping out the wall, cleaning and treating that area with something that will likk the mold and inhibit it from regrowing. AFTER the source of the leak is found and repaired.