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This paper proposes an h p -version Legendre spectral collocation method for solving third-kind VIEs with nonlinear vanishing delay, which effectively addresses the singularity of the equation and the intricate impact of the delay term. The solvability of the algebraic system is established, and the convergence of the h p -version spectral collocation method under the L ∞ -norm is proved. Numerical experiments are performed to validate our theoretical predictions. [ABSTRACT FROM AUTHOR]