It sounds like a nonlinear fit problem. Here I generate some random points using the starting expression n + (a*x+b)^c and then fit the data using the target expression (m*x+n)^k:
data = With[{a = 1, b = 2, n0 = 3, c = 3/4},
Table[{x, n0 + (a*x + b)^c}, {x, RandomReal[{0, 5}, {10}]}]];
NonlinearModelFit[data, (m*x + n)^k, {m, n, k}, x]
