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

<sub>higher order terms are readily available, for example, the term of order $(\pi-\phi)^4$ between square brackets is $-\frac{8}{1440}\left(8 \left(3 (ka)^4+80 (ka)^2+15\right)+15 i \pi  ka \left(3 (ka)^2+2\right)\right)(\pi -\phi)^4$.</sub>