For weights $p(t)$ and $q(t)$ with a finite number of power-law-type singularities we obtain necessary and sufficient conditions for the inequality $$\|s_n^{(p)}(f)q\|_{L^\eta(-1,1)}\le C\|fq\|_{L^\eta(-1,1)},$$ to hold, where $s_n^{(p)}(f)$ is a partial sum of the Fourier series of the function $f$ in terms of polynomials orthogonal on $[-1,1]$ with weight $p(t)$. This inequality is used to solve the problem concerning convergence in the mean and also convergence almost everywhere of the partial sum $s_n^{(p)}(f)$.