If I read this correctly, you want the integration performed over the dummy variable z, with a limit from 0 to s. The list you import as "s" contains the values for s.

Here is code that would do that:

(* some made-up data *)
data = Range[1, 5]

(* {1,2,3,4,5} *)

(* h can be just an expression *)
h = Sqrt[0.3*((1 + z)^3) + 0.7];

c = 3*10^8;

(* a function that accepts s as an argument -- numerical integration *)
r[s_] := NIntegrate[c/h, {z, 0, s}]

(* applied to a single value *)
r[2]

(* 3.628413612813334`*^8 *)

(* mapped onto a list of values for s *)    
r /@ data

(* {2.31428*10^8, 3.62841*10^8, 4.45206*10^8, 5.02273*10^8, 
 5.44653*10^8} *)