Wiener-Hopf plus Hankel operators $W(a)+H(b):L^p(\mathbb{R}^+)\to L^p(\mathbb{R}^+)$ with generating functions $a$ and $b$ from a subalgebra of $L^\infty(\mathbb{R})$ containing almost periodic functions and Fourier images of $L^1(\mathbb{R})$-functions are studied. For $a$ and $b$ satisfying the so-called matching condition $$ a(t)a(-t)=b(t)b(-t),\quad tı\mathbb{R}, $$ we single out some classes of operators $W(a)+H(b)$ which are subject to the Coburn-Simonenko theorem.