Kitchen Extension and waste pipe

You say it's a temporary stud wall, I think it is meant to be permanent. I would remove the plasterboard then pull the flush or run water and see what causes the noise.The void should be stuffed with sound insulation quilt anyway. Or get the contractor to do it as it is part of the job.

Copper pipe not clipped to stud wall Possible water hammer

It's probably the plastic waste expanding and contracting then rubbing against the clips or structure. Open it up and have a look-see.

Most likely the copper pipes are not properly clipped to the wall. Extra clippers will definitely help to stop the noise . Cavity should be filled with insulation too.

The void behind the plasterboard needs to be filled with loose fill insulation by making an access at the top of the wall pour the loose fill and fill the void fro the top. How do you know there are copper pipes behind the void running along the soil stack pipe. If there are copper pipes running then they should be clipped down. Another point does the clicking sound happen when the water is being used or when the waste water is being discharge ie flushing the toilet ? As it could be the water pipes are tapping each other when the water is being used need to do a process of elimination the reason why you need to use easy fill is that you will not need to remove all the plasterboard in order to fill the void I’m assuming all decoration has been done hope that helps Note: clicking is it constant click click click does is stop after some time ?