Consider the i'th largest element, and a particular permutation. It'll inserted into the k-sized heap if it appears before no more than k-1 of the (i - 1) larger elements in the permutation.
The probability of that heap-insertion happening is 1 if i <= k, and k/i if i > k.
From this, you can compute the expectation of the number of heap adjustments, using linearity of expectation. It's sum(i = 1 to k)1 + sum(i = k+1 to n)k/i = k + sum(i = k+1 to n)k/i = k * (1 + H(n) - H(k)), where H(n) is the n'th harmonic number.
This is approximately k log(n) (for k << n), and you can compute your average cost from there.