This paper studies a class of singular integral equations with two convolution kernels and Hilbert kernels in the function class $L^2[-\mathtt{\pi },\mathtt{\pi }]$. This study use discrete Fourier transform, generalized residue theorem and related theory of linear algebra to transform the singular integral equation containing the mixture of Hilbert kernel and convolution kernel into a system of linear equations. Finally, this paper obtains the general solution and solvable conditions of the equation on $L^2[-\mathtt{\pi },\mathtt{\pi }]$.