Summary: We established the existence of a positive solution of nonlinear fractional differential equations \(\mathfrak L(D)[x(t) - x(0)] = f(t, x_t), t \in (0, b]\) with finite delay \(x(t) = \omega(t), t \in [-\tau, 0]\), where \(lim_{t \rightarrow 0} f(t, x_t) = +\infty\), that is, \(f\) is singular at \(t = 0\) and \(x_t \in C([-\tau, 0], \mathbb R^{\geq 0}\). The operator of \(\mathfrak L(D)\) involves the Riemann-Liouville fractional derivatives. In this problem, the initial conditions with fractional order and some relations among them were considered. The analysis rely on the alternative of the Leray-Schauder fixed point theorem, the Banach fixed point theorem, and the Arzela-Ascoli theorem in a cone.