I am following David Tong's notes on QFT. On page 58, he applies Wick's theorem to $\psi\psi\rightarrow\psi\psi$ scattering for a scalar field with a Yukwara interaction term. The claim is, that by Wick's theorem, the only contribution to $$\int d^4x_1d^4x_2 T[\psi^\dagger(x_1)\psi(x_1)\phi(x_1)\psi^\dagger(x_2)\psi(x_2)\phi(x_2)]\tag{3.46}$$ is $$:\psi^\dagger(x_1)\psi(x_1)\psi^\dagger(x_2)\psi(x_2):\Delta_F(x_1,x_2).\tag{3.47}$$ That is, contracting on the two $\phi$ terms. However, I do not see why we don't get a contribution by contracting on a $\psi^\dagger$ and $\psi$, i.e, a contribution from
$$:\psi^\dagger(x_1)\psi(x_2):\Delta_F(x_1, x_2)\Delta_F(x_1,x_2).$$
when the one of incoming and outgoing momenta is the same. That is to say, if the initial state is $|pq>$ and the final state is $<pq|$, why do we not get a further second order contribution.
Indeed, if I try it, I get
$$<pq|:\psi^\dagger(x_1)\psi(x_2):|pq>$$ $$<0|2\sqrt{E_pE_q}b_{p}b_{q}^\dagger e^{i(-q.x_1+p.x_2)}|0>,$$ so
$$<pq|:\psi^\dagger(x_1)\psi(x_2)\Delta_F(x_1, x_2)\Delta_F(x_1,x_2):|pq>$$ $$=<p|\int \frac{d^4k}{2\pi^2} \frac{\delta(p-q)}{(k^2-m^2)((p-k)^2-m^2)}|q>,$$
which certainly gives a non-zero contribution when $p=q$.