Rewrite an expression in a specific form

Questions
1

Hello. I am relatively new to Mathematica so I am not completely aware of all it’s able to do.
I have an expression that looks as follows:

n + (a*x+b)^c

Where ‘a’, ‘b’, ‘n’, and ‘c’ are constant numbers, and the value of ‘c’ is between 0.5 and 1.5.
I would like rewrite this expression such that it will look as follows:

(m*x+n)^k

The final expression can be an approximation of the first expression.

2

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]