This article investigates the existence and stability of solutions for a class of three-point boundary value problems involving Riemann-Liouville fractional derivatives of order less than one. We considered nonlinear fractional differential equations subject to nonlocal boundary conditions that include a singular condition at the origin and a global fractional condition at interior points. Our approach generalizes previous work, allowing for singular behavior near zero, and employs the Leray-Schauder fixed point theorem to establish the existence of bounded solutions without requiring contractive conditions on the nonlinear term. Instead, we imposed a local integrability condition known as the $L^p$-Carathéodory condition. Furthermore, we studied the Ulam-Hyers-Rassias stability of approximate solutions by means of a Gronwall-type inequality adapted to weakly singular kernels. Two concrete examples were also included to illustrate the theory.