Installing a cleanout for 4" corrugated black pipe foundation drainage
The problem with this approach is that using a sewer auger in the corrugated drain can actually damage the soft plastic drain tile. The cutting edges on the auger can tear into the pipe and then allow the collapse of the drain. The corrugated pipe is used because it is cheap and quick to install. Solid PVC pipe with holes is a better choice if you want to be able to clean the drain. I would use a garden hose to flush the system from the clean outs you install. Splice in the wye style style Tees with a rigid PVC riser near as many corners as you can, and keep flushing from one corner to the next. You will need multiple access points because you will not be able to push the hose around the corners.