The authors consider the functional-differential equation \[ u''(t)= f(u)(t),\tag{1} \] where \(f: C^2_{\text{loc}}(a,b)\to L_{\text{loc}}(a,b)\) is a continuous operator which may be singular, i.e., when \(f(u)(\cdot)\not\in L[a, b]\) for some \(u\in C^1_{\text{loc}}(a, b)\). A special case of (1) is the differential equation with deviating arguments \[ u''(t)= f_0(t, u(\tau_1(t)), u(\tau_2(t))),\tag{2} \] where \(f_0: (a,b)\times\mathbb{R}^2\to \mathbb{R}\) satisfies the local Carathéodory conditions and \(\tau_i(a, b)\to(a, b)\), \(i= 1, 2\), are continuous functions. The singular equations (1) and (2) are discussed together with the two-point boundary conditions \[ u(a_+)= 0,\quad u(b_-)= 0,\tag{3} \] \[ u(a_+)= 0,\quad u'(b_-)= 0\tag{3} \] and with the additional condition \[ \int^b_a u^{\prime 2}(t)\,dt< \infty.\tag{5} \] Sufficient conditions for the existence and uniqueness of solutions (in the class \(\text{AC}^1_{\text{loc}}(a, b))\) to problems \((j)\), (3), (5) and \((j)\), (4), (5), \(j= 1, 2\), are presented. Examples show that some conditions are unimprovable. Sufficient conditions use one-sided growth restriction on the operator \(f\) and the function \(f_0\). Existence results are proved by the principle of a priori boundedness.