This paper considers the randomly weighted sums generated by some dependent subexponential primary random variables and some arbitrarily dependent random weights. To study the randomly weighted sums with infinitely many terms, we establish a Kesten-type upper bound for their tail probabilities in presence of subexponential primary random variables and under a certain dependence among them. Our result extends the study of Chen [5] to the dependent case. As applications, we derive some asymptotic formulas for the tail probability and the Value-at-Risk of total aggregate loss in a multivariate operational risk cell model.