This paper achieves the weakly asymptotic formulas of the large deviations for the discounted aggregate claims in a time-dependent risk model with widely upper orthant dependent and dominatedly-varying-tailed claims, where the time-dependence structure is defined by a conditional tail probability of the claim size given the inter-arrival time before the claim. Further, if the claims are consistently varying tailed or regularly varying tailed, some asymptotic formulas of the large deviations are established