We know that computing a maximum flow resp. a minimum cut of a network with capacities is equivalent; cf. the max-flow min-cut theorem.
We have (more or less efficient) algorithms for computing maximum flows, and computing a minimum cut given a maximum flow is neither hard nor expensive, either.
But what about the reverse? Given a minimum cut, how can we determine a maximum flow? Without solving Max-Flow from scratch, of course, and preferably faster than that, too.
From the minimum cut, we know the maximum flow value. I don't see how this information helps the standard approaches augmenting-path and push-relabel, although adapting the latter seems slightly more plausible.
We can not use the minimum cut to split the network in two parts and recurse since that won't shrink the problem in the worst case (if one partition is a singleton); also we would not have a minimum cut of the smaller instances.
Does knowing the value of the maximum flow speed up solving the Max-Flow LP, maybe via the complementary slackness conditions?