 Solution

In one chiral sector there are three derivatives $\partial X$. A nonzero Wick contraction can pair two derivatives with each other and contract the remaining derivative with an exponential; this gives a metric times one momentum and hence the terms linear in $k$. Alternatively, all three derivatives can contract with exponential operators, giving three momenta and the term proportional to ${\alpha }^{\prime }{k}^3$. The same alternatives occur independently in the antiholomorphic sector.