a = Sum[x[r] ((Sum[x[i], {i, n}] - x[r])^2 - Sum[x[i]^2, {i, n}]), {r, n}];

    Module[{t0, t1},
     rule = Sum[p1___  x[a_]^i__  p2___, {a_, n}] :> p1 p2 s[i];
     a /. x[k_] -> t0 x[k]^t1 
       /. Sum[a__, l_List] :> (Sum[#, l] & /@ Expand@a) 
       //. rule /.  t0 -> 1 /. t1 -> 1]
     
    (*s[1]^3 - 3 s[1] s[2] + s[3]*)