You have defined x to be a function of Pn only, but in fact, it is dependent on several other parameters and you are not passing those values into the function. So, they are global. But inside Manipulate, they are local control variables. One way to address this is to make sure all of your functions pass in all of the parameter values:

x[Pn_, h_, d_] := (h (d/Pn)^d)^(1/(1 - d));

nx[Pn_, h_, d_] := d/Pn x[Pn, h, d];

Bx[Pn_, h_, d_] := (1 - d)*x[Pn, h, d];

f[z_] := -0.0003*z^3 + 0.006*z^2 - 0.1*z + 1;

n[Pn_, k_, h_, d_] := k f[x[Pn, h, d]];

Bn[Pn_, k_, h_, d_] := Pn n[Pn, k, h, d];

r[Pn_, k_, h_, d_] := Bx[Pn, h, d] + Bn[Pn, k, h, d];

xc[Pn_, a_, k_, h_, d_] := a r[Pn, k, h, d];

nc[Pn_, a_, k_, h_, d_] := (1 - a)/Pn r[Pn, k, h, d];

eq[Pn_, k_, a_, h_, d_] := 
  n[Pn, k, h, d] - nc[Pn, a, k, h, d] - nx[Pn, h, d];

Manipulate[
 FindRoot[eq[Pn, k, a, h, d] == 0, {Pn, 0.5966337871664869`}], {h, 4, 
  5}, {d, 0.3, 2}, {a, 0.5, 1}, {k, 20, 100}]

You could try to assign the outputs of the Simplify[] commands to variables, for instance out6, out7, etc. (to keep things looking the same).

thank you very much, the last thing I would like to know is if I can somehow make it show me the values of Manipulate [] as tables.

Manipulate[
 FindRoot[{eq[Pn, k, a, h, d] == 0, 
   x[Pn, h, d] == casax}, {{casax, 1}, {Pn, 0.5966337871664869}}], {h,
   4, 5}, {d, 0.3, 2}, {a, 0.5, 1}, {k, 20, 100}]

the problem is that I have not managed to run me otherwise, everything works perfectly but fails to work me without using the OUT [%] also would eliminate all that and the program work without placing me Simplify[]

Thx