Let $ξ_1, ξ_2,\ldots$ be a sequence of independent and identically distributed random variables with zero mean, finite second moment and regularly varying right distribution tail. Motivated by a stop-loss insurance model, we consider a threshold sequence $M_n\gg (n\ln n)^{1/2},$ $n\to \infty,$ and establish the asymptotics of the probabilities of the large deviations of the form $ \sum_{j=1}^n(ξ_j \wedge M_n)>x$ in the whole spectrum of $x$-values in the region $O(M_n).$ The asymptotic representations for these probabilities obey the "multiple large jumps principle" and have different forms in the vicinities of the multiples $kM_n$ of the censoring threshold values, on the one hand, and inside intervals of the form $((k-1)M_n, kM_n),$ on the other. We show that there is a "smooth transition" of these representations from one to the other when the deviation $x$ increases to a multiple of $M_n$, "crosses" it and then moves away from it.