How to change the outputs of the neural network that the error function receives

Given a neural network with 2 output classes there are 2 ways of assigning its real-valued output, M in [0,1], to binary classes. You have one target class, if M>0.5 assign to class 1. Else, do not (equivalent to assign to class 2). You have two target classes. If M_{1} > M_{2} assign to class 1, else assign to class 2. I would like my output to be biased s.t. if M>0.8 assign to class 1, else class 2.Incidentally, (this is not the main question) would this be equivalent to If 0.2*M_{1} >0.8*M_{2} assign to class 1, else class 2. I can change the final outputs in this way. However, I am running the network through some loops and I would like my changes to be recognised within the network's architecture. It is right that the error function minimises some sort of distance (perhaps least squared) between the output, M, and the target, y? In my case it will be minimising the distance between the old, unmodified output, rather than my threshold adjusted output. Do you know how I can tell the error function about this new output? Many thanks for any help