<p>We aim to address the initial value problem of a kind of two-term nonlinear fractional differential equation that exhibits singular behavior. Our approach involves precisely characterizing the solution’s singular properties through a psi-series approximation and developing two efficient Chebyshev collocation methods by eliminating the initial and potential blow-up singularities. We first reformulate the equation into a Volterra integral equation of the second kind and develop a successive approximation method to derive the psi-series solution close to the origin. The psi-series solution serves two crucial purposes: it approaches the solution near the origin and helps identify the blow-up time based on Darboux’s theorem regarding the blow-up problem. After addressing the initial and potentially terminal singularities, we discretize the equivalent Hadamard finite-part and Volterra integral equations over the regular interval and propose two singularity-removing Chebyshev spectral collocation methods. We provide a detailed discussion on the construction and implementation of the method for the hyper-singular integral form. We also establish the error estimate of the method for the Volterra integral form. Finally, we present several examples to illustrate the effectiveness of the psi-series solution in characterizing the singular behavior and highlight the high accuracy achieved through the singularity-removing Chebyshev collocation methods.</p>