With version 9.0.1,

    f[x_] := (p^2 + k^2 - 2 p k x)/(x - (p^2 + k^2 + 1 - ((p^2 - k^2)^2)/4)/(2 p k)); 
    ans9 = Integrate[f[x], {x, -1, 1}, PrincipalValue -> True]
    (* ConditionalExpression[1/2 k p (-8 + ((-4 + (k^2 - p^2)^2) 
       ArcCoth[(8 k p)/(-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))])/(k p)), 
       k^4 + p^4 < 4 + 4 p^2 + 2 k^2 (2 + p^2) && (k - p)^2 (-2 + k + p) (2 + k + p) > 4] *)

Some insight can be gained by plotting the solution ad its region of validity as specified by `ConditionalExpression`,

    RegionPlot[Evaluate[ans9[[2]]], {k, -5, 5}, {p, -5, 5}, FrameLabel -> {k, p}]

![enter image description here][1]

    Plot3D[Evaluate[ans9[[1]]], {k, -5, 5}, {p, -5, 5}, PlotPoints -> 100,
      PlotRange -> All, Mesh -> None, AxesLabel -> {k, p, z}]


![enter image description here][2]

Thus, the solution is real and continuous over a wider range than that given by the `ConditionalExpression`.  Nonetheless, the actual range is bounded by singularities, so it is not surprising that the second `Integrate` in the question does not yield an answer.  Of course, even without the singularities, `Integrate` might fail, if no known symbolic solution exists.

Because, I presume, `k` and `p` are meant to be `Reals`, it is reasonable to inform `Integrate` of this.  

    ans9r = Integrate[f[x], {x, -1, 1}, PrincipalValue -> True, 
      Assumptions -> k ∈ Reals && p ∈ Reals]
    (* ConditionalExpression[1/2 k p (-8 + ((-4 + (k^2 - p^2)^2) 
       ArcTanh[(8 k p)/(-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))])/(k p)),
       (-4 + (k - p)^2) (k + p)^2 <= 4 && (k - p)^2 (-2 + k + p) (2 + k + p) <= 4] *)

which produces a different but related solution (perhaps, a different branch).  

![enter image description here][3]

![enter image description here][4]

Between them, the two solutions appear to cover all of `p - k` space.  Nonetheless, it seems unlikely that patching the two together and then integrating would be successful.

Version 10.1 produces symbolic solutions that look different.  

    (* ConditionalExpression[1/4 (-(-4 + (k^2 - p^2)^2) Log[-k p] + (k^2 - p^2)^2 (Log[k p]
       - Log[4 - (-4 + (k - p)^2) (k + p)^2] + Log[4 - (k - p)^2 (-2 + k + p) (2 + k + p)])
       - 4 (4 k p + Log[k p] - Log[4 - (-4 + (k - p)^2) (k + p)^2] + 
       Log[4 - (k - p)^2 (-2 + k + p) (2 + k + p)])),
       (-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))/(k p) ∈ Reals] *)

and

    (* ConditionalExpression[1/4 (-16 k p - (-4 + (k^2 - p^2)^2) 
       Log[4 - (-4 + (k - p)^2) (k + p)^2] + (-4 + (k^2 - p^2)^2) 
       Log[4 - (k - p)^2 (-2 + k + p) (2 + k + p)]), 
       k^4 + p^4 <= 4 + 4 p^2 + 2 k^2 (2 + p^2) && (-4 + (k - p)^2) (k + p)^2 <= 4] *)

Their 3D plots appear identical respectively the fourth and second plots above, although their asserted ranges of validity differ.  Note that a straightforward application of `FullSimplify` does not succeed in showing them to be identical.

  [1]: https://i.sstatic.net/fEb8I.png
  [2]: https://i.sstatic.net/7l6bw.png
  [3]: https://i.sstatic.net/aWC7F.png
  [4]: https://i.sstatic.net/GKJTR.png