The controls in a Manipulate are localized. Changing the value of the controls for g, w or d updates only the references to g, w or d that explicitly appears in the first argument to Manipulate. No other updates are done.

An example that works, a Manipulate plotting your function n:

n[x_, g_, w_, d_] := Sqrt[-1/8 (-1 + Sqrt[1 - d^2]) (2 g^2 + w^2 - w^2 Cos[2 g * x])]

Manipulate[
    Plot[n[t, g, w, d], {t, 0, 10}],
    {{g, 1}, 0, 20, Appearance -> "Labeled"},
    {{w, 1}, 0, 2 Pi, Appearance -> "Labeled"},
    {{d, 0.5}, 0, 1, Appearance -> "Labeled"}
 ]

Thank you for your answer.

I think the problem that I have lies within the computation of the Integral of n[x]:k[x]. I need k[x] to plot the Graph but there are some parameters in k[x] (g and w) that I would like to be able to manipulate because I want to look at this time evolution for as many different parameter sets as possible. But I cannot compute the general Integral k[x] without specifying g and w in advance because the NIntegrate option gives me errors for the unspecified parameters. I plotted the whole thing for the parameters {g,w,d}={1,1,1/2}) but I don't want to manually change all the parameters in all the calculations every time.

I need to plot this situation for many different Parameter sets of g,w and d so I thought about using a program where I only need to specify my 3 parameters in advance and all the steps necessary to get to the plot were taken without me having to manually change the parameters all the time. Am I right in the assumption that the Manipulate option can do this?

It does not necessarily have to be Manipulate, if there is an Option like specifying a table of sets of parameters and the plot is then made for all of them I would be happy too.