Q.nb

ClearAll[hi1, h1, g1]
hi1 = Compile[{{x, _Real}, {En, _Real}, {pz, _Real}, {k4, _Real}, \
{kp, _Real}},
  ((Sqrt[0.316/(1. + 
            1.2*((k4 + 0.5*En)^2 + kp + (x*pz)^2))^1.*0.316/(1. + 
            1.2*((k4 - 0.5*En)^2 + kp + ((1. - x)*pz)^2))^1.])*((1. - 
          x)*0.316/(1. + 1.2*((k4 + 0.5*En)^2 + kp + (x*pz)^2))^1. + 
       x*0.316/(1. + 
            1.2*((k4 - 0.5*En)^2 + 
               kp + ((1. - x)*pz)^2))^1.))/(((k4 + 0.5*En)^2 + 
       kp + (x*pz)^2 + (0.316/(1. + 
             1.2*((k4 + 0.5*En)^2 + kp + (x*pz)^2))^1.)^2)*((k4 - 
          0.5*En)^2 + 
       kp + ((1. - x)*
          pz)^2 + (0.316/(1. + 
             1.2*((k4 - 0.5*En)^2 + kp + ((1. - x)*pz)^2))^1.)^2))
  ]

h1[x_?NumericQ, En_?NumericQ, pz_?NumericQ] :=
 
 1./(En^2 + pz^2 + 0.24^2)*NIntegrate[
   hi1[x, En, pz, k4, kp], {k4, -\[Infinity], \[Infinity]}, {kp, 
    0, \[Infinity]},
   Method -> "LocalAdaptive", 
   MaxRecursion -> 
    100];(*LocalAdaptive seems to work slightly faster*)

g1[x_] := h1[0.5, x, 2.];
Timing@Total[Table[g1[x], {x, N@Range[-2, 2, 1/8]}]]