The outgoing photon attaches to the electron line before the incoming photon, so the virtual electron carries momentum $k = p - q'$.
The computational chain
From diagram to number—every step of the amplitude calculation.
The diagram
The outgoing photon is emitted at vertex 1, then the incoming photon is absorbed at vertex 2. The virtual electron carries momentum $k = p - q'$ with $k^2 = (p - q')^2 = u$.
Feynman rules expression
$$i\mathcal{M}_u = (-ie)^2\,\epsilon_\mu(q)\,\epsilon_\nu^*(q')\,\bar{u}(p')\gamma^\mu\frac{i(\not{p}-\not{q}'+m)}{(p-q')^2-m^2}\gamma^\nu u(p)$$
Reading along the fermion line from right to left: the outgoing photon vertex $\gamma^\nu$ acts first, then the propagator with momentum $p - q'$, then the incoming photon vertex $\gamma^\mu$.
Squared amplitude (u-channel alone)
$$\overline{|\mathcal{M}_u|^2} \supset -2e^4\frac{m^2 - s}{m^2 - u}$$
The electron mass regulates the propagator denominator $m^2 - u$ and cannot be dropped.