I slightly disagree...
In most cases you will still lose the cutting if it's developing mold, however, I have had cases where I take the cutting out of the mix I use from Jon's Uline bag method and could clearly see where part of the cutting was rotten and part was not. I then cut off all of the rotten part, usually until I got to wood that still looked viable and didn't have any brown in it (brown in the inner part of the wood usually means that area has already begun rotting). So I cut it back until it looks like it does when you fresh cut it off the tree with green, etc. Then I redo the whole soapy water and bleach process and restart it. In some cases (very few), the remaining piece survived and grew. However, in most of those cases, it is because I caught the rot early and didn't have to remove too much of the cutting, leaving plenty of stored energy to grow.
I've also had cuttings with rot on one end still put out roots and was able to remove rotten part and plant the rest and it grew. These cases are extremely rare.
Figpig, If the bark is mushy and rubs off with your finger your cutting is already rotten and a goner. Check soil moisture, probably too damp.