Series expansion of the integrand around $\varphi=\pi$ and integration gives
$$H = 2ika\int_{-\pi/2}^{\pi/2}\cos{(\varphi-\phi)}e^{ika[\cos{\varphi}+\cos{(\varphi-\phi)}]}\ d\varphi$$
$$\qquad\qquad=2ika\left[-2+\frac{1}{12} \left(4 (ka)^2+3 i \pi  ka+12\right) (\pi -\phi)^2+{\cal O}(\pi-\phi)^4\right].$$