In this paper, we present estimates for solutions and for the attraction domain of the trivial solution for systems with delayed and nonlinear weighted homogeneous right-hand side of positive degree. The results are achieved via a generalization of the Lyapunov-Krasovskii functional construction presented recently for homogeneous systems with standard dilation. Along with the classical approach for the calculation of the estimates within the Lyapunov-Krasovskii framework, we develop a novel approach which combines the use of Lyapunov- Krasovskii functionals with ideas of the Razumikhin framework. More precisely, a lower bound for the functional on a special set of functions inspired by the Razumikhin condition is constructed, and an additional condition imposed on the solution of the comparison equation ensures that this bound can be used to estimate all solutions in a certain neighbourhood of the trivial one. An example shows that this approach yields less conservative estimates in comparison with the classical one.