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We analyze chainlike N-soliton dynamics in a weakly nonlocal, essentially nonintegrable system described by the cubic-quintic nonlinear Schrödinger equation. Quintic nonlinearity is not assumed to be small. This system is reduced to a generalized complex Toda chain model. Numerical simulations demonstrate adverse action of both cubic and quintic nonlocal responses, in their own right, on the quasi-equidistant train propagation, with a development of a chaotic regime. From the Toda chain model, we predict a possibility of mutually compensating both types of nonlocality-induced distortion, restoring thereby a deterministic mode of the train propagation in a weakly nonlocal medium. Analytical predictions corroborate well with numerical results. [ABSTRACT FROM AUTHOR]